# Invertible elements in associates and semigroups. 1

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Quasigroups and Related Systems 5 (1998), Invertible elements in associates and semigroups. 1 Fedir Sokhatsky Abstract Some invertibility criteria of an element in associates, in particular in n-ary semigroups, are given. As a corollary, axiomatics for polyagroups and n-ary groups are obtained. Invertible elements play a special role in the theory of n-ary groupoids. For example, the structure of operations in an associate without invertible elements is still open. However, in the associate of the type (r, s, n) the structure of its operation is determined by Theorem 4 from [2] as soon as there exists at least one r-multiple invertible element in it. In particular, this theorem reduces the study of the groupoid to the study of associate of the type (1, s, n) with invertible elements. Since, as was shown in [3], a binary semigroup with an invertible element is exactly a monoid, so we will take the characteristic to introduce a notion of multiary monoid. 1. Necessary informations Let (Q; f) be an (n + 1)-ary groupoid. The operation f and the groupoid (Q; f) are called (i, j)-associative, if the identity f(x 0,..., x i 1, f(x i,..., x i+n ), x i+n+1,..., x 2n ) = f(x 0,..., x j 1, f(x j,..., x j+n ), x j+n+1,..., x 2n ) Mathematics Subject Classication: 20N15 Keywords: polyagroup, invertible element, n-ary group

2 54 F. Sokhatsky holds in (G; f). Denition 1. A groupoid (Q; f) of the arity n + 1 is said to be an associate of the type (r, s, n), where r divides s, s divides n, and n > s, if it is (i, j)-associative for all (i, j) such, that i j 0 (mod r), and i j (mod s). In an associate of the type (s, n), that is of the type (1, s, n), the number s will be called a degree of associativity, and the associative operation f will be called s-associative. The least of the associativity degrees will be called a period of associativity. The following theorem is proved in [4]. Theorem 1. Let (Q; f) be an associate of a type (r, s, n). If the words w 1 and w 2 dier from each other by bracketting only; the coordinate of every f's occurrence in the words w 1 and w 2 is divisible by r and there exists an one-to-one correspondence between f's occurrences in the word w 1 and those in the word w 2 such that the corresponding coordinates are congruent modulo s, then the formula w 1 = w 2 is an identity in (Q; f). Here the coordinate of the i-th occurrence of the symbol f in a word w is called a number of all individual variables and constants, appearing in the word w from the beginning of w to the i-th occurrence of the operation symbol f. To dene an invertible element we need the notion of a shift. Let (Q; f) be an (n + 1)-ary groupoid. The notation i a denotes a sequence a,..., a (i times). A transformation λ i,a of the set Q, which is determined by the equality λ i,a (x) = f( i a, x, n i a ), (1) is said to be an i-th shift of the groupoid (Q; f), induced by an element a. Hence, the i-th shift is a partial case of the translation (see [1]). If an i-th shift is a substitution of the set Q, then the element a is called i-invertible. If an element a is i-invertible for all i multiple of r, then it is called r-multiple invertible, when r = 1 it is called invertible. The unit is always invertible, since, it determines a shift being an identity transformation.

3 Invertible elements in associates 55 The notion of an invertible element for binary and n-ary groupoids coincides with a well known one. Namely, if (Q; ) is a semigroup, and a is its arbitrary invertible element, that is the shifts λ 0,a and λ 1,a are substitutions of the set Q, then it is easy to prove (see [3]), that the elements 0,a(a), 1,a(a) are right and left identity elements in the semigroup. Therefore, 0,a(a) = 1,a(a) is an identity element, left and right inverse elements of the element a are λ 2 1,a(a) and λ 2 0,a(a) respectively. Thus, a 1 := λ 2 1,a(a) = λ 2 0,a(a) is an inverse element of a. If an element a of a multiary groupoid is i-invertible, then the element i,a (a) coincides with the i-th skew element of a, which is denoted by ā i, where ā = ā 0, and it is determined by the equality f( i a, ā i, n i a ) = a. The following two lemmas are proved in [2] Lemma 2. If in an associate of the type (r, s, n) an element a is s- multiple invertible and i 0 (mod s), then there exists a unique i-th skew of the element a, and, in addition, the equality ā = ā i (2) holds. Lemma 3. In every associate of the type (r, s, n) for every s-multiple invertible element of the element a and for all i 0 (mod s) the following identities are true f( i a, ā, n i 1 a, x) = x, f(x, n i 1 a, ā, i a) = x. (3) 2. Criteria of invertibility of elements One of the main results of this article is the following. Theorem 4. An element a Q is r-multiple invertible in an associate (Q; f) of the type (r, s, n) i there exists an element ā Q such that f(ā, a,... a, x) = x, f(x, a,... a, ā) = x (4) holds for all x Q.

4 56 F. Sokhatsky Proof. If an element a is r-multiple invertible in (Q; f), then the relation (4) follows from (3) when i = 0. Let the relationship (4) hold. To establish the invertibility of the element a, we have to prove the existence of an inverse transformation for every of the shifts induced by the element a. This follows from the following lemma. Lemma 5. Let (Q; f) be an associate of the type (r, s, n). If for a Q there exists an element ā satisfying (4), then every i-th shift, induced by a, has an inverse transformation, which can be found by the formulae ( ) 0,a(x) = f x, n s 1 a, ā, ( ) λn,a(x) 1 = f ā, s 1 a, ā, n s 1 a, x, ( i,a (x) = f n s i ) a, x, i 1 a, ā, when 0 < i n s, ( ) i,a (x) = f ā, n i 1 a, x, s 1 a, ā, i s a, when s i < n. Proof of Lemma. If i in (3) is a multiple of s, then x (4) = f(ā, n 1 a, x) (4) = f(ā, i 1 a, f( n a, ā), n i 1 a, x) = f(f(ā, a), n i 1 a, ā, n i 1 a, x) (4) = f( a, i ā, n i 1 a, x). The other relationships from (3) are proved by the same way: x (4) = f(x, n 1 a, ā) (4) = f(x, n i 1 a, f(ā, a), n i 1 a, ā) = f(x, n i 1 a, ā, i 1 a, f( a, n ā)) (4) = f(x, n i 1 a, ā, a). i Let us prove that the transformation 0,a, which is determined by the equality (5) is inverse to λ 0,a. 0,aλ 0,a (x) (5) = f(λ n,a (x), n s 1 a, ā) (1) = f(f(x, a), n n s 1 a, ā) = f(x, n s 1 a, f( a, n ā), s 1 = f(x, n 1 = x, (5) λ 0,a 0,a(x) (5) = λ 0,a f(x, n s 1 a, ā) (1) = f(f(x, n s 1 a, ā), a) n = f(x, n s 1 a, f(ā, a)) n (3) = f(x, n s 1 a, ā, a) s (3) = x.

5 Invertible elements in associates 57 Hence λ 0,a 0,a = 0,aλ 0,a = ε, where ε is the identity mapping. Thus, the transformation 0,a, determined by the equality (5), is inverse to the shift λ 0,a. Analogously one can prove the other equalities from (5). λn,a n,a (x) (5) = f(ā, s 1 a, ā, n s 1 a, λ n,a (x)) (1) = f(ā, s 1 a, ā, n s 1 a, f( a, n x)) = f(ā, s 1 a, f(ā, n a), n s 1 a, x) (3) = f(ā, n 1 a, x) (3) = x, λ n,a n,a(x) (5) = λ n,a f(ā, s 1 a, ā, n s 1 a, x) (1) = f( a, n f(ā, s 1 a, ā, n s 1 a, x)) = f(f( n a, ā), s 1 a, ā, n s 1 a, x) (3) = f( s a, ā, n s 1 a, x) (3) = x. Let number i n 3 be a multiple of r. Then λ i,a (5) i,a (x) = f( a, i f( n s i a, x, i 1 a, ā), n i a ) (3) = f( a, i f( n s i a, x, i 1 a, ā), n i 1 a, f( a, n ā)) = f(f( n s a, x), i 1 a, f(ā, n a), n i 1 a, ā) = (3) = f(f( n s a, x), n 1 = f(x, n 1 = x. If i = n s, then the equality (5) denes the transformation which implies n s,a(x) = f(ā, s 1 a, x, n s 1 a, ā), n s,aλ n s,a (x) (1) = f(ā, s 1 a, f( n s a, x, s a), n s 1 a, ā) If i < n s, then = f(f(ā, n 1 a, x), n 1 = f(x, n 1 = x. i,a λ i,a(x) (5) = f( n s i a, f( a, i x, n i a ), i 1 a, ā) (3) = f(f(ā, a), n n s i 1 a, f( a, i x, n i a ), i 1 a, ā) = f(f(ā, n i 1 a, f( n s a, ā, a), s i 1 a, x), n 1 = x.

6 58 F. Sokhatsky If i s, then λ i,a (5) i,a (x) = f( a, i f(ā, n i 1 a, x, s 1 a, ā, i s a ), n i a ) (3) = f(f(ā, a), n i 1 a, f(ā, n i 1 a, x, s 1 a, ā, i s a ), n i a ) = f(f(ā, i 1 a, f( a, n ā), n i 1 a, x), s 1 a, ā, n s a ) (3) = x. To prove i,a λ i,a(x) = ε, we consider two cases: i = s and i > s. If i = s, then (5) can be rewritten as Therefore we get If i > s, then s,aλ s,a (x) (1) = f(ā, n s 1 a, f( a, s x, n s a ), s 1 a, ā) s,a(x) = f(ā, n s 1 a, x, s 1 a, ā). = f(f(ā, n 1 a, x), n 1 = f(x, n 1 = x. i,a λ i,a(x) (5) = f(ā, n i 1 a, f( a, i x, n i a ), s 1 a, ā, i s a ) (3) = f(ā, n i 1 a, f( a, i x, n i a ), s 1 a, ā, i s 1 a, f( a, n ā)) = f(f(ā, n 1 a, x), n i 1 a, f( a, s ā, n s a ), i 1 = x. The lemma and the theorem has been proved. Since for r = s = 1 we obtain an (n + 1)-ary semigroup, then the following corollary is true. Corollary 1. An element a Q is invertible in an (n + 1)-ary semigroup (Q; f) i there exists an element ā Q such that (4) holds for all x Q. 3. Monoids and invertible elements In the associate of the type (r, s, n) the structure of its operation is determined by Theorem 4 from [2] as soon as there exists at least one r-multiple invertible element in it. In particular, this theorem implies

7 Invertible elements in associates 59 (see Corollary 11 in [2]) that the study of the groupoid reduces to the study of an associate of the type (1, s, n) with invertible elements, that is why we will consider the last ones only. Since, as it was shown above, a binary semigroup with an invertible element is exactly a monoid, so we will use this characteristic to introduce its generalization and we will call it invert (a multiary monoid was called a semigroup with an identity element). Since every invertible element of an invert determines some decomposition monoid, natural questions on the relations between the algebraic notions for a monoid and decomposition monoids as well as about relations between dierent decompositions of the same monoid arise. Here we will consider this relation between the sets of invertible elements. Denition 2. An associate of the type (1, s, n) containing at least one invertible element will be called an invert of the type (s, n). When s = 1, then an invert is an (n + 1)-ary semigroup containing at least one invertible element. So, every (n + 1)-ary monoid is an invert. If an invert has at least one neutral element e, then, as follows from the results given below, the automorphism of its e-decomposition is identical, therefore its associativity period is equal to one, that is, such invert is a monoid. Every (n + 1)-ary group is an invert, since every its element is invertible. The next statement, which follows from Theorem 4 in [2], gives a decomposition of the operation of an invert. Theorem 6. Let (Q; f) be an (n + 1)-ary invert of the associativity period s. Then for every its invertible element 0 there exists a unique triple of operations (+, ϕ, a) such, that (Q; +) is a semigroup with a neutral element 0, an automorphism ϕ and an invertible element a, which satises the following relations: ϕ n (x) + a = a + x, ϕ s (a) = a, (6) f(x 0, x 1,..., x n ) = x 0 + ϕ(x 1 ) + ϕ 2 (x 2 ) + + ϕ n (x n ) + a. (7) And conversely, if an endomorphism ϕ and an element a of an semigroup (Q; +) are connected by the relations (6), then the groupoid

8 60 F. Sokhatsky (G; f) determined by the equality (7) is an (n + 1)-ary associate of the associativity degree s. We will use the following terminology: (Q; +) is called a monoid of the 0-decomposition; ϕ is said to be an automorphism of the 0- decomposition; a is called a free member of the 0-decomposition; +, ϕ, a are called components of the 0-decomposition; and (Q; +, ϕ, a) is said to be an algebra of the 0-decomposition of the invert (Q; f). Lemma 7. Let k be a nonnegative integer, which is not greater than n and is a multiple of s, 0 is an arbitrary invertible element of the invert (Q; f), then the components of its 0-decomposition are uniquely determined by the following equalities x + y = f(x, k 1 0, 0, n k 1 0, y); a = f(0, 0,..., 0); a = 0; ϕ i (x) = 0,0λ i,0 (x) = f( i 0, x, n i 1 0, 0); ϕ i (x) = n,0λ n i,0 (x) = f( 0, n i 1 0, x, i 0) for all i = 1,..., n 1. Proof. In [2] the rst three of the equalities were proved. Since n divides s, then (6) implies ϕ n (a) = a, therefore ϕ n ( 0) = ϕ n ( a) = ϕ n (a) = a = 0. The transformation ϕ is an automorphism of the semigroup (Q; f), therefore ϕ(0) = 0 and ϕ i (x) = ϕ i (x) a + a = 0 + ϕ(0) + + ϕ i 1 (0) + ϕ i (x)+ +ϕ i+1 (0) + + ϕ n 1 (0) + ϕ n ( 0) + a (6) = f( i 0, x, n i 1 0, 0). Let us now make use of the relationships (5): 0,0λ i,0 (x) (5) = f(f( i 0, x, n i 0 ), n s 1 0, 0, s 1 0, 0) = f( i 0, x, n i 1 0, f( n s 0, 0, s 1 0, 0)) (3) = f( i 0, x, n i 1 0, 0) (7) = ϕ i (x). (8)

9 Invertible elements in associates 61 n,0λ n i,0 (x) (5) = f( 0, s 1 0, f( n i 0, x, i 0)) = = f(f( 0, s 1 0, 0, n s 0 ), n i 1 0, x, 0) i (3) = f( 0, n i 1 0, x, 0) i (7) = 0 + ϕ(0) + + ϕ n i (x) + ϕ n i+1 (0) + + ϕ n (0) + a (6) = a + ϕ n (ϕ i (x)) + a (6) = ϕ i (x). The lemma is proved. Corollary 2. Under the notations of Theorem 6 the associativity period of the invert is equal to the least of the numbers s, such that ϕ s (a) = a, i.e. it is equal to the length of the orbit of the element a, when we consider the action of the cyclic group ϕ generated by the automorphism ϕ. If an element x is invertible in an a-decomposition monoid, then its inverse element will be denoted by x a or by x 1 a depending on additive or multiplicative notation of the a-decomposition monoid. It should be noted that the element x a is uniquely determined by the elements a and x. Theorem 8. An element of an invert will be invertible i it is invertible in one (hence, in every) of the decomposition monoids. Proof. Let (Q; f) be an invert of the type (s, n) with an invertible element 0 and (Q; +) be a 0-decomposition monoid. Let x be invertible in (Q; f) and let x 0 := f(0, n s 1 x, x, s 1 x, 0). (9) To prove that the element x 0 is inverse to x, we will use the equality (8) when k = s. x + ( x 0 ) (8) = f(x, s 1 0, x 0 ) (9) = f(x, s 1 0, f(0, n s 1 x, x, s 1 x, 0)) = f(f(x, s 1 0, 0, n s 0 ), n s 1 x, x, s 1 x, 0) (3) = f( n s x, x, s 1 x, 0) (3) = 0.

10 62 F. Sokhatsky x 0 + x (8) = f( x 0, s 1 0, x) (9) = f(f(0, n s 1 x, x, s 1 x, 0), s 1 0, x) = f(0, n s 1 x, x, s 1 x, f( 0, s 0, n s 1 0, x)) (3) = f(0, n s 1 x, x, x) s (3) = 0. Hence, the element x 0 is inverse to x in (Q; +). Conversely, let the element x be invertible in the 0-decomposition monoid (Q; +). Then the element f(0, x,..., x, 0) (7) = ϕx + ϕ 2 x + + ϕ n 1 x + a is invertible in (Q; +) too. Let us dene the element x by In particular, this means that x = f(0, x,... x, 0) 0. (10) x + f(0, x,... x, 0) 0 = 0. Then for any element y of Q we get the following relations: y = 0 + y = x + f(0, n 1 x, 0) + y (8) = f(f( x, s 1 0, f(0, n 1 x, 0)), s 1 0, y) = f(f( x, s 1 0, 0, n s 0 ), n 1 x, f( s 0, 0, n s 1 0, y)) (3) = f( x, n 1 x, y), y = y + 0 (10) = y + f(0, n 1 x, 0) + x ) (8) = f (f(y, s 1 0, f(0, n 2 x, 0)), s 1 0, x ) = f (f(y, s 1 0 ), n 2 x, f( 0, s 0, n s 1 (3) 0, x) = f(y, n 2 x, x). From Theorem 4 we get the invertibility of the element x in the invert (Q; f). Corollary 3. The sets of all invertible elements of multiary monoid and decomposition monoids are pairwise equal.

11 Invertible elements in associates 63 Corollary 4. Let 0 be an invertible element of a monoid (Q; f) of the type (s, n) and let k be multiple of s. Then the element x will be invertible in (Q; f) i there exists an element x 0 such that f(x, s 1 0, x 0 ) = f( x 0, s 1 0, x) = 0 (11) hold. Proof. The equality (11) according to the equalities (8) means the truth of the relations x+( x 0 ) = x 0 +x = 0, where (Q; +) is the 0-decomposition monoid, that is the element x is invertible in (Q; +). Hence, by Theorem 8 it will be invertible in the associate (Q; f). Lemma 9. Let (Q; f) be an invert of the type (s, n), (+, ϕ, a) be its 0-decomposition. A triple (, ψ, b) of operations dened on Q will be a decomposition of (Q; f) i there exists an invertible in (Q; +) element e satisfying the conditions x y = x e + y, ψ(x) = e + ϕ(x) ϕ(e), b = e + ϕ(e) + ϕ 2 (e) + + ϕ n (e) + a. (12) The algebra (Q;, ψ, b) in this case will be e-decomposition of the invert (Q; f). Proof. Let (, ψ, b) be e-decomposition of the invert (Q; f), then x y (8) = f(x, s 1 e, ē, n s 1 e, y) (7) = x + ϕe + + ϕ s 1 (e)+ +ϕ s (ē) + ϕ s+1 (e) + + ϕ n 1 (e) + ϕ n (y) + a (6) = x + (ϕ(e) + + ϕ s 1 (e) + ϕ 3 (ē)+ +ϕ s+1 (e) + + ϕ n 1 (e) + a) + y. Hence x y = x + c + y for some c Q and all x, y Q. In particular, when x = e, y = 0 and x = 0, y = e we get the invertibility of the element c in (Q; +), and the relation c = e. Next, ψ(x) (8) = f(e, x, n 2 e, ē) (7) = e + ϕ(x) + ϕ 2 (e) + + +ϕ n 1 (e) + ϕ n (ē) + a = e + ϕ(x) + d

12 64 F. Sokhatsky for some d Q. But e = ψ(e) = e + ϕ(e) + d, therefore d = ϕ(e). On the other hand, let an element e be invertible in (Q; +) and determine a triple of operations (, ψ, b) on Q by the equalities (12). The invertibility of the element e in the invert (Q; f) is ensured by Theorem 8. If the component of the e-decomposition of the invert (Q; f) are denoted by (, χ, c), then just proved assertion gives x y = x e + y, χ(x) = e + ϕ(x) ϕ(e), c = e + ϕ(e) + ϕ 2 (e) + + ϕ n (e) + a. Therefore (, ψ, b) = (, χ, c). This means, that (, ψ, b) will be a decomposition of (Q; f). We say that the monoids (Q; ) and (Q; +) dier from each other by a unit, if the equality x y = x e + y holds for some invertible in (Q; +) element e, because they coincide once their units coincide. This relationship between monoids is stronger than isomorphism since the translations L 1 e and Re 1 are isomorphic mappings from one to the other. Therefore the following statement is obvious. Corollary 5. Any two decomposition monoids of the same invert differ from each other by a unit. Theorem 10. The set of all invertible elements of an invert is its subquasigroup and coincides with the group of all invertible elements of any of its decomposition monoids. Proof. Let (Q; f) be an invert of the type (s, n) and let (+, ϕ, a) be its 0-decomposition. Theorem 8 implies that the sets of all invertible elements of groupoids (Q; f) and (Q; +) coincide. Denote this set by G. Inasmuch as G is a subgroup of the monoid (Q; +) and ϕg = G, a G, so for any elements c 0, c 1,..., c n G the element f(c 0, c 1,..., c n ) (7) = c 0 + ϕ(c 1 ) + ϕ 2 (c 2 ) + + ϕ n (c n ) + a is in G also. Furthermore for any number i = 0, 1,..., n the solution of f(c 0,..., c i 1, x, c i+1,..., c n ) = c, where c G, is unique and

13 Invertible elements in associates 65 coincides with the element ( x (6) = ϕ i ϕ i 1 (c i 1 ) ϕ(c 1 ) c 0 + c a ) ϕ n (c n ) ϕ i+1 (c i+1 ). which is in G too. Hence, (G; f) is a subquasigroup of (Q; f). (13) Theorem 11. The period of associativity of the invert determined by (θ, a) coincides with the number of dierent skew elements of an invertible element and with the length of the orbit < θ > (a), where < θ > is the automorphism group generated by θ. Proof. Let number s be the period of associativity of the (n + 1)-ary invert (Q; f) and let x be any of its invertible elements. Denote by (, ψ, b) the x-decomposition of the invert (Q; f). Since f( x, i ψ n i ( x), n i x ) (8) = f( x, i f( n i x, x, i 1 x, x), n i 1 x, f( x, n x)) = f(f( n x, x), i 1 x, f( x, n x), n i 1 x, x) (3) = f( n x, x) = x, then the i-th skew x i of the element x is determined by the equality x i = ψ n i ( x) (8) = f( n i x, x, i 1 x, x), i = 0, 1,..., n. (14) Inasmuch as, in accordance with the equality (8), ψ s ( x) = ψ s (b 1 ) = (ψ s (b)) 1 = b 1 = x, there are at most s dierent skew elements of the element x: Namely x, x 1,..., x s 1. Suppose, for some numbers i, j with i < j < s, the i-th and j-th skew elements of x coincide. The results obtained imply the equality ψ n i ( x) = ψ n j ( x), so that ψ j i ( x) = x. The last equality together with equality ψ s ( x) = x give the relation ψ d ( x) = x, where d = g.c.d.(s, j i). In view of (8) this implies ψ d (b) = b. It follows from Theorem 6 that the pair (d, n) will be a type of the invert (Q; f). At the same time d < s. A contradiction to the denition of the associativity period.

14 66 F. Sokhatsky Thus, the element x has exactly s skew elements. They are determined by the relation (14) and by any full collection of pairwise noncongruent indices modulo s. Corollary 6. If one of skew elements of an invertible element x of a monoid coincides with x, then all skew elements of x are equal and this invert is a semigroup. Proof. Let 0 i = 0. The relation (14) implies ϕ n i ( 0) = 0, where ϕ denotes an automorphism of the 0-decomposition. Apply to the last equality ϕ i we obtain ϕ n ( 0) = ϕ i (0). Since ϕ n ( 0) = ϕ n ( a) = ϕ n (a) = a = 0, then accounting to (8) we obtain f( 0, i 0, n i 1 0, 0) = 0, that is 0 = 0. Thus a = f(0,..., 0) = 0 and ϕ(a) = ϕ(0) = 0 = a. Hence, by Theorem 11 the associativity period of the invert is equal to 1, i.e. the invert is a semigroup. Corollary 7. An invert of associativity period s has at least s + 1 dierent invertible elements. Corollary 8. An invert having at most two invertible elements is associative i.e. is a semigroup. Proof. If an invert has exactly one invertible element, then it will be associative by Corollary 6, since its skews coincide with it. If the invert has exactly two invertible elements a and b, then ā 0 = a or ā 0 = b. If ā 0 = a, then according to Corollary 6 the invert is a semigroup. If ā 0 = b, then Theorem 11 implies ā 1 = a. And Theorem 11 implies that the invert is a quasigroup. Axiomatics of polyagroups Both for binary and for n-ary cases an associative quasigroup is called a group. Therefore, retaining this regularity we will introduce the notion of a polyagroup.

15 Invertible elements in associates 67 Denition 3. When s < n the s-associative (n + 1)-ary quasigroup we will call a nonsingular polyagroup of the type (s, n). It is easy to see, that s = 1 means the polyagroup is an (n + 1)-ary group. Theorem 6 implies an analogue of Gluskin-Hosszú theorem. Proposition 12. Any polyagroup of the type (s, n) is (i, j)-associative for all i, j with i j (mod s). When s is its associativity period, then no other (i, j)-associativity identity holds. Theorem 13. Let (Q; f) be an associate of the type (s, n) and s < n, n > 1. Then the following statements are equivalent 1) (Q; f) is a polyagroup, 2) every element of the associate is invertible, 3) for every x Q there exists x Q such that f( x, x,..., x, y) = f(y, x,..., x, x) = y (15) holds for all y Q, 4) (Q; f) has an invertible element 0 and for every x Q there exists y Q such that f(x, s 1 0, y) = 0, f(y, s 1 0, x) = 0 (16) holds. Proof. 1) 2) follows from Theorem 10; 2) 3) from Corollary 1; 2) 4) from Corollary 4. When s = 1 we get a criterion for n-ary groups. Corollary 9. Let (Q; f) be (n + 1)-ary a semigroup. Then the following statements are equivalent 1) (Q; f) is an (n + 1)-ary group, 2) every element of the semigroup is invertible, 3) for every x Q there exists x Q such that (15) holds for every y Q, 4) (Q; f) has an invertible element 0 and for every x Q there exists y Q such that (16) hold.

16 68 F. Sokhatsky References [1] V. D. Belousov: n-ary quasigroups, (Russian) Chi³in u, tiinµa [2] F. N. Sokhatsky: About associativity of multiary operations, (Russian) Diskretnaya matematika 4 (1992), [3] F. Sokhatsky: On superassociative group isotopes, Quasigroups and Related Systems 2 (1995), [4] F. M. Sokhatsky: On associativity of multiplace operations, Quasigroups and Related Systems 4 (1997), Received May 24, 1998 and in revised form October 11, 1998 Vinnytsia State Pedagogical University Vinnytsia Ukraine

### I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

### Solutions to TOPICS IN ALGEBRA I.N. HERSTEIN. Part II: Group Theory

Solutions to TOPICS IN ALGEBRA I.N. HERSTEIN Part II: Group Theory No rights reserved. Any part of this work can be reproduced or transmitted in any form or by any means. Version: 1.1 Release: Jan 2013

### 2.1 Functions. 2.1 J.A.Beachy 1. from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair

2.1 J.A.Beachy 1 2.1 Functions from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair 21. The Vertical Line Test from calculus says that a curve in the xy-plane

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### MODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction.

MODULAR ARITHMETIC 1 Working With Integers The usual arithmetic operations of addition, subtraction and multiplication can be performed on integers, and the result is always another integer Division, on

### We give a basic overview of the mathematical background required for this course.

1 Background We give a basic overview of the mathematical background required for this course. 1.1 Set Theory We introduce some concepts from naive set theory (as opposed to axiomatic set theory). The

### CHAPTER 5. Number Theory. 1. Integers and Division. Discussion

CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers a and b we say a divides b if there is an integer c such that b = ac. If a divides b, we write a

### Math 223 Abstract Algebra Lecture Notes

Math 223 Abstract Algebra Lecture Notes Steven Tschantz Spring 2001 (Apr. 23 version) Preamble These notes are intended to supplement the lectures and make up for the lack of a textbook for the course

### 4. Expanding dynamical systems

4.1. Metric definition. 4. Expanding dynamical systems Definition 4.1. Let X be a compact metric space. A map f : X X is said to be expanding if there exist ɛ > 0 and L > 1 such that d(f(x), f(y)) Ld(x,

### GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.

Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the

### Chapter 7. Continuity

Chapter 7 Continuity There are many processes and eects that depends on certain set of variables in such a way that a small change in these variables acts as small change in the process. Changes of this

### Degree Hypergroupoids Associated with Hypergraphs

Filomat 8:1 (014), 119 19 DOI 10.98/FIL1401119F Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Degree Hypergroupoids Associated

### UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure

UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress

### Logic, Algebra and Truth Degrees 2008. Siena. A characterization of rst order rational Pavelka's logic

Logic, Algebra and Truth Degrees 2008 September 8-11, 2008 Siena A characterization of rst order rational Pavelka's logic Xavier Caicedo Universidad de los Andes, Bogota Under appropriate formulations,

### Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson

Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement

### it is easy to see that α = a

21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore

### SUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by

SUBGROUPS OF CYCLIC GROUPS KEITH CONRAD 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by g = {g k : k Z}. If G = g, then G itself is cyclic, with g as a generator. Examples

### Notes on Determinant

ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without

### MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

### Let H and J be as in the above lemma. The result of the lemma shows that the integral

Let and be as in the above lemma. The result of the lemma shows that the integral ( f(x, y)dy) dx is well defined; we denote it by f(x, y)dydx. By symmetry, also the integral ( f(x, y)dx) dy is well defined;

### Introduction to Finite Fields (cont.)

Chapter 6 Introduction to Finite Fields (cont.) 6.1 Recall Theorem. Z m is a field m is a prime number. Theorem (Subfield Isomorphic to Z p ). Every finite field has the order of a power of a prime number

### A simple criterion on degree sequences of graphs

Discrete Applied Mathematics 156 (2008) 3513 3517 Contents lists available at ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam Note A simple criterion on degree

### DETERMINANTS. b 2. x 2

DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in

### A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries

Acta Math. Univ. Comenianae Vol. LXVI, 2(1997), pp. 285 291 285 A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE E. T. BASKORO, M. MILLER and J. ŠIRÁŇ Abstract. It is well known that Moore digraphs do

### ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS

ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS John A. Beachy Northern Illinois University 2014 ii J.A.Beachy This is a supplement to Abstract Algebra, Third Edition by John A. Beachy and William D. Blair

### 1 = (a 0 + b 0 α) 2 + + (a m 1 + b m 1 α) 2. for certain elements a 0,..., a m 1, b 0,..., b m 1 of F. Multiplying out, we obtain

Notes on real-closed fields These notes develop the algebraic background needed to understand the model theory of real-closed fields. To understand these notes, a standard graduate course in algebra is

### = 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that

Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without

### A New Generic Digital Signature Algorithm

Groups Complex. Cryptol.? (????), 1 16 DOI 10.1515/GCC.????.??? de Gruyter???? A New Generic Digital Signature Algorithm Jennifer Seberry, Vinhbuu To and Dongvu Tonien Abstract. In this paper, we study

### 1 Sets and Set Notation.

LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most

### Galois Theory III. 3.1. Splitting fields.

Galois Theory III. 3.1. Splitting fields. We know how to construct a field extension L of a given field K where a given irreducible polynomial P (X) K[X] has a root. We need a field extension of K where

### Correspondence analysis for strong three-valued logic

Correspondence analysis for strong three-valued logic A. Tamminga abstract. I apply Kooi and Tamminga s (2012) idea of correspondence analysis for many-valued logics to strong three-valued logic (K 3 ).

### Cartesian Products and Relations

Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) :(a A) and (b B)}. The following points are worth special

### F. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)

Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p -CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p

### 3. Applications of Number Theory

3. APPLICATIONS OF NUMBER THEORY 163 3. Applications of Number Theory 3.1. Representation of Integers. Theorem 3.1.1. Given an integer b > 1, every positive integer n can be expresses uniquely as n = a

### Labeling outerplanar graphs with maximum degree three

Labeling outerplanar graphs with maximum degree three Xiangwen Li 1 and Sanming Zhou 2 1 Department of Mathematics Huazhong Normal University, Wuhan 430079, China 2 Department of Mathematics and Statistics

### Practice with Proofs

Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using

### Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

### SEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS. Nickolay Khadzhiivanov, Nedyalko Nenov

Serdica Math. J. 30 (2004), 95 102 SEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS Nickolay Khadzhiivanov, Nedyalko Nenov Communicated by V. Drensky Abstract. Let Γ(M) where M V (G) be the set of all vertices

### On the generation of elliptic curves with 16 rational torsion points by Pythagorean triples

On the generation of elliptic curves with 16 rational torsion points by Pythagorean triples Brian Hilley Boston College MT695 Honors Seminar March 3, 2006 1 Introduction 1.1 Mazur s Theorem Let C be a

### Chapter 7. Homotopy. 7.1 Basic concepts of homotopy. Example: z dz. z dz = but

Chapter 7 Homotopy 7. Basic concepts of homotopy Example: but γ z dz = γ z dz γ 2 z dz γ 3 z dz. Why? The domain of /z is C 0}. We can deform γ continuously into γ 2 without leaving C 0}. Intuitively,

### DEGREES OF ORDERS ON TORSION-FREE ABELIAN GROUPS

DEGREES OF ORDERS ON TORSION-FREE ABELIAN GROUPS ASHER M. KACH, KAREN LANGE, AND REED SOLOMON Abstract. We construct two computable presentations of computable torsion-free abelian groups, one of isomorphism

### fg = f g. 3.1.1. Ideals. An ideal of R is a nonempty k-subspace I R closed under multiplication by elements of R:

30 3. RINGS, IDEALS, AND GRÖBNER BASES 3.1. Polynomial rings and ideals The main object of study in this section is a polynomial ring in a finite number of variables R = k[x 1,..., x n ], where k is an

### University of Lille I PC first year list of exercises n 7. Review

University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients

GENERATING SETS KEITH CONRAD 1 Introduction In R n, every vector can be written as a unique linear combination of the standard basis e 1,, e n A notion weaker than a basis is a spanning set: a set of vectors

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### Turing Degrees and Definability of the Jump. Theodore A. Slaman. University of California, Berkeley. CJuly, 2005

Turing Degrees and Definability of the Jump Theodore A. Slaman University of California, Berkeley CJuly, 2005 Outline Lecture 1 Forcing in arithmetic Coding and decoding theorems Automorphisms of countable

### ON GALOIS REALIZATIONS OF THE 2-COVERABLE SYMMETRIC AND ALTERNATING GROUPS

ON GALOIS REALIZATIONS OF THE 2-COVERABLE SYMMETRIC AND ALTERNATING GROUPS DANIEL RABAYEV AND JACK SONN Abstract. Let f(x) be a monic polynomial in Z[x] with no rational roots but with roots in Q p for

### minimal polyonomial Example

Minimal Polynomials Definition Let α be an element in GF(p e ). We call the monic polynomial of smallest degree which has coefficients in GF(p) and α as a root, the minimal polyonomial of α. Example: We

### No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics

No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results

### DEFINABLE TYPES IN PRESBURGER ARITHMETIC

DEFINABLE TYPES IN PRESBURGER ARITHMETIC GABRIEL CONANT Abstract. We consider the first order theory of (Z, +,

### 3. Equivalence Relations. Discussion

3. EQUIVALENCE RELATIONS 33 3. Equivalence Relations 3.1. Definition of an Equivalence Relations. Definition 3.1.1. A relation R on a set A is an equivalence relation if and only if R is reflexive, symmetric,

### About the inverse football pool problem for 9 games 1

Seventh International Workshop on Optimal Codes and Related Topics September 6-1, 013, Albena, Bulgaria pp. 15-133 About the inverse football pool problem for 9 games 1 Emil Kolev Tsonka Baicheva Institute

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22 RAVI VAKIL CONTENTS 1. Discrete valuation rings: Dimension 1 Noetherian regular local rings 1 Last day, we discussed the Zariski tangent space, and saw that it

### The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992), 33 37) Bart de Smit

The fundamental group of the Hawaiian earring is not free Bart de Smit The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992),

### Computability Theory

CSC 438F/2404F Notes (S. Cook and T. Pitassi) Fall, 2014 Computability Theory This section is partly inspired by the material in A Course in Mathematical Logic by Bell and Machover, Chap 6, sections 1-10.

### POLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS

POLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS RUSS WOODROOFE 1. Unique Factorization Domains Throughout the following, we think of R as sitting inside R[x] as the constant polynomials (of degree 0).

### FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

### Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize

### Geometric Transformations

Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted

### U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

### Section 2.7 One-to-One Functions and Their Inverses

Section. One-to-One Functions and Their Inverses One-to-One Functions HORIZONTAL LINE TEST: A function is one-to-one if and only if no horizontal line intersects its graph more than once. EXAMPLES: 1.

### MA2C03 Mathematics School of Mathematics, Trinity College Hilary Term 2016 Lecture 59 (April 1, 2016) David R. Wilkins

MA2C03 Mathematics School of Mathematics, Trinity College Hilary Term 2016 Lecture 59 (April 1, 2016) David R. Wilkins The RSA encryption scheme works as follows. In order to establish the necessary public

### Categoricity transfer in Simple Finitary Abstract Elementary Classes

Categoricity transfer in Simple Finitary Abstract Elementary Classes Tapani Hyttinen and Meeri Kesälä August 15, 2008 Abstract We continue to study nitary abstract elementary classes, dened in [7]. We

### Group Fundamentals. Chapter 1. 1.1 Groups and Subgroups. 1.1.1 Definition

Chapter 1 Group Fundamentals 1.1 Groups and Subgroups 1.1.1 Definition A group is a nonempty set G on which there is defined a binary operation (a, b) ab satisfying the following properties. Closure: If

### Math 319 Problem Set #3 Solution 21 February 2002

Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod

### The Mean Value Theorem

The Mean Value Theorem THEOREM (The Extreme Value Theorem): If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers

### Applications of Fermat s Little Theorem and Congruences

Applications of Fermat s Little Theorem and Congruences Definition: Let m be a positive integer. Then integers a and b are congruent modulo m, denoted by a b mod m, if m (a b). Example: 3 1 mod 2, 6 4

### Introducing Functions

Functions 1 Introducing Functions A function f from a set A to a set B, written f : A B, is a relation f A B such that every element of A is related to one element of B; in logical notation 1. (a, b 1

### 2.3 Convex Constrained Optimization Problems

42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

### The Division Algorithm for Polynomials Handout Monday March 5, 2012

The Division Algorithm for Polynomials Handout Monday March 5, 0 Let F be a field (such as R, Q, C, or F p for some prime p. This will allow us to divide by any nonzero scalar. (For some of the following,

### CRITICAL ALGEBRAS AND THE FRATTINI CONGRUENCE. Emil W. Kiss and Samuel M. Vovsi

CRITICAL ALGEBRAS AND THE FRATTINI CONGRUENCE Emil W. Kiss and Samuel M. Vovsi Abstract. In this note we show that if two critical algebras generate the same congruencepermutable variety, then the varieties

### Group Theory. Contents

Group Theory Contents Chapter 1: Review... 2 Chapter 2: Permutation Groups and Group Actions... 3 Orbits and Transitivity... 6 Specific Actions The Right regular and coset actions... 8 The Conjugation

### GROUPS ACTING ON A SET

GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule for

### Our goal first will be to define a product measure on A 1 A 2.

1. Tensor product of measures and Fubini theorem. Let (A j, Ω j, µ j ), j = 1, 2, be two measure spaces. Recall that the new σ -algebra A 1 A 2 with the unit element is the σ -algebra generated by the

### Local periods and binary partial words: An algorithm

Local periods and binary partial words: An algorithm F. Blanchet-Sadri and Ajay Chriscoe Department of Mathematical Sciences University of North Carolina P.O. Box 26170 Greensboro, NC 27402 6170, USA E-mail:

### v w is orthogonal to both v and w. the three vectors v, w and v w form a right-handed set of vectors.

3. Cross product Definition 3.1. Let v and w be two vectors in R 3. The cross product of v and w, denoted v w, is the vector defined as follows: the length of v w is the area of the parallelogram with

### Lecture 13 - Basic Number Theory.

Lecture 13 - Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that A divides B, denoted

### CODING TRUE ARITHMETIC IN THE MEDVEDEV AND MUCHNIK DEGREES

CODING TRUE ARITHMETIC IN THE MEDVEDEV AND MUCHNIK DEGREES PAUL SHAFER Abstract. We prove that the first-order theory of the Medvedev degrees, the first-order theory of the Muchnik degrees, and the third-order

### Elements of Abstract Group Theory

Chapter 2 Elements of Abstract Group Theory Mathematics is a game played according to certain simple rules with meaningless marks on paper. David Hilbert The importance of symmetry in physics, and for

### TOPIC 4: DERIVATIVES

TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the

### Linear Maps. Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007)

MAT067 University of California, Davis Winter 2007 Linear Maps Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007) As we have discussed in the lecture on What is Linear Algebra? one of

### INTRODUCTORY SET THEORY

M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,

### Factoring Polynomials

Factoring Polynomials Sue Geller June 19, 2006 Factoring polynomials over the rational numbers, real numbers, and complex numbers has long been a standard topic of high school algebra. With the advent

### Lecture Notes: Matrix Inverse. 1 Inverse Definition

Lecture Notes: Matrix Inverse Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk Inverse Definition We use I to represent identity matrices,

### Mathematics of Cryptography

CHAPTER 2 Mathematics of Cryptography Part I: Modular Arithmetic, Congruence, and Matrices Objectives This chapter is intended to prepare the reader for the next few chapters in cryptography. The chapter

### Galois representations with open image

Galois representations with open image Ralph Greenberg University of Washington Seattle, Washington, USA May 7th, 2011 Introduction This talk will be about representations of the absolute Galois group

### Lecture 1: Schur s Unitary Triangularization Theorem

Lecture 1: Schur s Unitary Triangularization Theorem This lecture introduces the notion of unitary equivalence and presents Schur s theorem and some of its consequences It roughly corresponds to Sections

Chapter 4 Complementary Sets Of Systems Of Congruences Proceedings NCUR VII. è1993è, Vol. II, pp. 793í796. Jeærey F. Gold Department of Mathematics, Department of Physics University of Utah Don H. Tucker

### Linear Algebra Notes for Marsden and Tromba Vector Calculus

Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of

University of Oslo MAT2 Project The Banach-Tarski Paradox Author: Fredrik Meyer Supervisor: Nadia S. Larsen Abstract In its weak form, the Banach-Tarski paradox states that for any ball in R, it is possible

### Research Article On the Rank of Elliptic Curves in Elementary Cubic Extensions

Numbers Volume 2015, Article ID 501629, 4 pages http://dx.doi.org/10.1155/2015/501629 Research Article On the Rank of Elliptic Curves in Elementary Cubic Extensions Rintaro Kozuma College of International

### Math 231b Lecture 35. G. Quick

Math 231b Lecture 35 G. Quick 35. Lecture 35: Sphere bundles and the Adams conjecture 35.1. Sphere bundles. Let X be a connected finite cell complex. We saw that the J-homomorphism could be defined by

### Review for Final Exam

Review for Final Exam Note: Warning, this is probably not exhaustive and probably does contain typos (which I d like to hear about), but represents a review of most of the material covered in Chapters

### The Dirichlet Unit Theorem

Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

### Inverses and powers: Rules of Matrix Arithmetic

Contents 1 Inverses and powers: Rules of Matrix Arithmetic 1.1 What about division of matrices? 1.2 Properties of the Inverse of a Matrix 1.2.1 Theorem (Uniqueness of Inverse) 1.2.2 Inverse Test 1.2.3

### Topic 1: Matrices and Systems of Linear Equations.

Topic 1: Matrices and Systems of Linear Equations Let us start with a review of some linear algebra concepts we have already learned, such as matrices, determinants, etc Also, we shall review the method

### *.I Zolotareff s Proof of Quadratic Reciprocity

*.I. ZOLOTAREFF S PROOF OF QUADRATIC RECIPROCITY 1 *.I Zolotareff s Proof of Quadratic Reciprocity This proof requires a fair amount of preparations on permutations and their signs. Most of the material

### CONTINUED FRACTIONS, PELL S EQUATION, AND TRANSCENDENTAL NUMBERS

CONTINUED FRACTIONS, PELL S EQUATION, AND TRANSCENDENTAL NUMBERS JEREMY BOOHER Continued fractions usually get short-changed at PROMYS, but they are interesting in their own right and useful in other areas