On loops with universal anti-automorphic inverse property
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1 On loops with universal anti-automorphic inverse property Department of Fundamental Mathematics Moldova State University Chisinau, Republic of Moldova The 5th International Mathematical Conference on Quasigroups and Loops (LOOPS 2015) Ohrid, Macedonia, June 30 - July 4, 2015
2 Middle Bol loops: Definition A loop (Q, ) is called a right (resp. left) Bol loop if it satisfies the identity (zx y)x = z(xy x), resp. x(y xz) = (x yx)z. Definition A loop (Q, ) is called a middle Bol loop if the identity (x y) 1 = y 1 x 1 (the anti-automorphic inverse property) is universal in (Q, ), i.e. if every loop isotope of (Q, ) satisfies the anti-automorphic inverse property.
3 Middle Bol loops: Definition A loop (Q, ) is called a right (resp. left) Bol loop if it satisfies the identity (zx y)x = z(xy x), resp. x(y xz) = (x yx)z. Definition A loop (Q, ) is called a middle Bol loop if the identity (x y) 1 = y 1 x 1 (the anti-automorphic inverse property) is universal in (Q, ), i.e. if every loop isotope of (Q, ) satisfies the anti-automorphic inverse property.
4 V. Belousov, around 1965 A loop (Q, ) is middle Bol if and only if the corresponding e-loop (Q,, /, \) satisfies the middle Bol identity: x (yz\x) = (x/z) (y\x). The middle Bol identity is invariant under loop isotopy and is a generalization of the Moufang identity: x(yz x) = xy zx. Every Moufang loop is a middle Bol loop. A middle Bol loop is a Moufang loop iff it is an LIP-loop (or iff it is an RIP-loop).
5 Middle Bol loops are power associative Proposition Every middle Bol loop (Q, ) satisfies the equality x, y Q, m, n Z. (y m //x 1 ) (x\\y n ) = y m+n, Corollary. If (Q, ) is a middle Bol loop then, for every x, y Q and every m, n Z, the following equality holds: y m y n = y m+n. Middle Bol loops are power associative (i.e. each element of (Q, ) generates an associative subloop).
6 Middle Bol loops are power associative Proposition Every middle Bol loop (Q, ) satisfies the equality x, y Q, m, n Z. (y m //x 1 ) (x\\y n ) = y m+n, Corollary. If (Q, ) is a middle Bol loop then, for every x, y Q and every m, n Z, the following equality holds: y m y n = y m+n. Middle Bol loops are power associative (i.e. each element of (Q, ) generates an associative subloop).
7 Middle Bol loops are isostrophes of right (left) Bol loops [Gwaramija, 1971] A loop (Q, ) is middle Bol iff there exists a right (left) Bol loop (Q, ), such that x y = (y xy 1 )y = y 1 \x, resp. x y = y(y 1 x y) = x/y 1, where / ( \ ) is the left (right) division in (Q, ) Remark Commutative middle Bol loops are isostrophes of Bruck loops, i.e. of (right) Bol loops with the automorphic inverse property (AIP): (x y) 1 = x 1 y 1, x, y Q.
8 Middle Bol loops are isostrophes of right (left) Bol loops [Gwaramija, 1971] A loop (Q, ) is middle Bol iff there exists a right (left) Bol loop (Q, ), such that x y = (y xy 1 )y = y 1 \x, resp. x y = y(y 1 x y) = x/y 1, where / ( \ ) is the left (right) division in (Q, ) Remark Commutative middle Bol loops are isostrophes of Bruck loops, i.e. of (right) Bol loops with the automorphic inverse property (AIP): (x y) 1 = x 1 y 1, x, y Q.
9 Some connections between middle Bol loops and the corresponding right (left) Bol loops If (Q, ) is a middle Bol loop and (Q, ) is the corresponding right (left) Bol loop then: 1. (Q, ) and (Q, ) have a common unit; 2. The left (right) inverse of an element x in (Q, ) coincide with the left (right) inverse of x in (Q, ); 3. If (Q, ) is the corresponding right Bol loop and (Q, ) is the corresponding left Bol loop for (Q, ), then x y = y x, x, y Q; 4. Two middle Bol loops are isotopic (isomorphic) iff the corresponding right (left) Bol loops are isotopic (resp. isomorphic); 5. If (L, ) is a subloop (normal subloop) of a finite left (right) Bol loop (Q, ) then (L, ) is a subloop (normal subloop) of the corresponding middle Bol loop (Q, ).
10 Commutants of middle Bol loops are subloops The commutant (centrum, commutative center, semicenter) of a loop (Q, ) is the set Remark C ( ) = {a Q a x = x a, x Q}. 1. The commutant C ( ) of an arbitrary loop (Q, ) is not always a subloop; 2. If C ( ) is a subloop of a loop (Q, ), then C ( ) is not always a normal subloop; 3. If (Q, ) is a Moufang loop then C ( ) is a subloop. But it is an open problem to characterize the Moufang loops for which the commutant is a normal subloop; 4. The commutant of a left (right) Bol loop is not always a subloop. For example, all 21 left Bol loops of order 16, with non-subloop commutants are constructed by Kinyon M. K. and Phillips J. D.(2004)
11 Commutants of middle Bol loops are subloops Let (Q, ) be a middle Bol loop and let (Q, ) be the corresponding right Bol loop. Denote by C ( ) (resp. by C ( ) ) the commutant of (Q, ) (resp. the commutant of (Q, )). Theorem [Syrbu P., Grecu I., 2014] Commutants of middle Bol loops are AIP-subloops (i.e. (x y) 1 = x 1 y 1, x, y Q). Corollary If (Q, ) is a middle Bol loop and (Q, ) is the corresponding right Bol loop then (C ( ), ) is a subloop of (Q, ).
12 Commutants of middle Bol loops are subloops Let (Q, ) be a middle Bol loop and let (Q, ) be the corresponding right Bol loop. Denote by C ( ) (resp. by C ( ) ) the commutant of (Q, ) (resp. the commutant of (Q, )). Theorem [Syrbu P., Grecu I., 2014] Commutants of middle Bol loops are AIP-subloops (i.e. (x y) 1 = x 1 y 1, x, y Q). Corollary If (Q, ) is a middle Bol loop and (Q, ) is the corresponding right Bol loop then (C ( ), ) is a subloop of (Q, ).
13 Commutants of middle Bol loops are subloops Let (Q, ) be a middle Bol loop and let (Q, ) be the corresponding right Bol loop. Denote by C ( ) (resp. by C ( ) ) the commutant of (Q, ) (resp. the commutant of (Q, )). Theorem [Syrbu P., Grecu I., 2014] Commutants of middle Bol loops are AIP-subloops (i.e. (x y) 1 = x 1 y 1, x, y Q). Corollary If (Q, ) is a middle Bol loop and (Q, ) is the corresponding right Bol loop then (C ( ), ) is a subloop of (Q, ).
14 A criterion for C ( ) = C ( ) Theorem Let (Q, ) be a middle Bol loop and let (Q, ) be the corresponding right Bol loop. Then C ( ) = C ( ) iff the following conditions hold: 1. x 1 xa = a, for a C ( ), and x Q; 2. (b x) x 1 = b, for b C ( ) and x Q. Corollary If C ( ) = C ( ) then a b = a b, a, b C ( ) = C ( ) ((C ( ), ) = (C ( ), )) is a commutative Moufang subloop. Remark We have no examples of middle Bol loops with not normal commutants.
15 A criterion for C ( ) = C ( ) Theorem Let (Q, ) be a middle Bol loop and let (Q, ) be the corresponding right Bol loop. Then C ( ) = C ( ) iff the following conditions hold: 1. x 1 xa = a, for a C ( ), and x Q; 2. (b x) x 1 = b, for b C ( ) and x Q. Corollary If C ( ) = C ( ) then a b = a b, a, b C ( ) = C ( ) ((C ( ), ) = (C ( ), )) is a commutative Moufang subloop. Remark We have no examples of middle Bol loops with not normal commutants.
16 A criterion for C ( ) = C ( ) Theorem Let (Q, ) be a middle Bol loop and let (Q, ) be the corresponding right Bol loop. Then C ( ) = C ( ) iff the following conditions hold: 1. x 1 xa = a, for a C ( ), and x Q; 2. (b x) x 1 = b, for b C ( ) and x Q. Corollary If C ( ) = C ( ) then a b = a b, a, b C ( ) = C ( ) ((C ( ), ) = (C ( ), )) is a commutative Moufang subloop. Remark We have no examples of middle Bol loops with not normal commutants.
17 The groups of regular substitutions in middle Bol loops Let (Q, ) be an arbitrary loop. Denote by L ( ) (R ( ), F ( ) ) - the group of left (resp. right, middle) regular substitutions of (Q, ), and by N ( ) l (N ( ) r, N ( ) m ) - the left (resp. right, middle) nucleus of (Q, ). Proposition If (Q, ) is a middle Bol loop and (Q, ) is the corresponding right Bol loop then the following equalities hold: 1. R ( ) = F ( ) = L ( ) ; 2. L ( ) = F ( ) ; 3. F( ) = R ( ); 4. L ( ) = IR ( ) I; 5. N ( ) r = N ( ) m = N ( ) l = N ( ) r ; 6. N ( ) l = N ( ) m, where F( ) is the group of adjoints of the middle regular substitutions of Q( ).
18 The groups of regular substitutions in middle Bol loops Let (Q, ) be an arbitrary loop. Denote by L ( ) (R ( ), F ( ) ) - the group of left (resp. right, middle) regular substitutions of (Q, ), and by N ( ) l (N ( ) r, N ( ) m ) - the left (resp. right, middle) nucleus of (Q, ). Proposition If (Q, ) is a middle Bol loop and (Q, ) is the corresponding right Bol loop then the following equalities hold: 1. R ( ) = F ( ) = L ( ) ; 2. L ( ) = F ( ) ; 3. F( ) = R ( ); 4. L ( ) = IR ( ) I; 5. N ( ) r = N ( ) m = N ( ) l = N ( ) r ; 6. N ( ) l = N ( ) m, where F( ) is the group of adjoints of the middle regular substitutions of Q( ).
19 Multiplications groups of a quasigroup Let (Q, ) be a quasigroup. Denote: LM(Q, ) =< L a a Q > - left multiplication group; RM(Q, ) =< R a a Q > - right multiplication group; M(Q, ) =< L a, R a a Q >- multiplication group, where L ( ) a (x) = a x, R ( ) a (x) = x a, a, x Q.
20 Connections between multiplication groups of a quasigroup and of its isostrophic loops Theorem Let (Q, ) be a quasigroup and let its isostrophe (Q, ), given by x y = ψ(y)\ϕ(x), x, y Q, where ϕ, ψ S Q, be a loop. The following statements hold: 1. LM(Q, ) =< I ( ) x ψ x Q >; 2. RM(Q, ) =< L ( ) x ϕ x Q >=< L ( ) 1 x L ( ) y x, y Q >; 3. M(Q, ) =< I ( ) x ψ, L ( ) < I ( ) x y ϕ x, y Q >= ψ, L ( ) 1 y L ( ) z x, y, z Q >; 4. If ϕ Aut(Q, ), then RM(Q, ) LM(Q, ); 5. LM(Q, ) =< RM(Q, ), ϕ >. Here I ( ) x : Q Q, I ( ) x (y) = y\x, x, y Q.
21 Connections between multiplication groups of a quasigroup and of its isostrophic loops Theorem Let (Q, ) be a quasigroup and let its isostrophe (Q, ), given by x y = ϕ(x)/ψ(y), x, y Q, where ϕ, ψ S Q, be a loop. The following statements hold: 1. LM(Q, ) =< I ( ) 1 x ψ x Q >; 2. RM(Q, ) =< R ( ) 1 x ϕ x Q >=< R ( ) 1 x R ( ) y x, y Q >; 3. M(Q, ) =< I ( ) 1 x ψ, R ( ) 1 y ϕ x, y Q >= < I ( ) 1 x ψ, R ( ) 1 y R ( ) z x, y, z Q >; 4. If ϕ Aut(Q, ), then RM(Q, ) RM(Q, ); 5. RM(Q, ) =< RM(Q, ), ϕ >.
22 Corollaries for isostrophic Bol loops Corollary 1 Let (Q, ) be a right Bol loop and let (Q, ) be the corresponding middle Bol loop. Then Corollary 2 RM(Q, ) = LM(Q, ). Let (Q, ) be a left Bol loop and let (Q, ) be the corresponding middle Bol loop. Then RM(Q, ) = RM(Q, ).
23 Corollaries for isostrophic Bol loops Corollary 1 Let (Q, ) be a right Bol loop and let (Q, ) be the corresponding middle Bol loop. Then Corollary 2 RM(Q, ) = LM(Q, ). Let (Q, ) be a left Bol loop and let (Q, ) be the corresponding middle Bol loop. Then RM(Q, ) = RM(Q, ).
24 Generalized regular mappings A mapping ϕ : Q Q is called a generalized left (right) regular mapping of (Q, ) if ϕ : Q Q, such that ϕ(x y) = ϕ (x) y (resp. ϕ(x y) = x ϕ (y)), for every x, y Q. In this case ϕ is called the adjoint of ϕ. Let (Q, ) be a quasigroup. We ll denote by R ( ) (L ( ) ) the group of all generalized right (left) regular mappings of (Q, ) and by R ( )(L ( )) the group of conjugates of the generalized right (left) regular mappings of (Q, ). C (Q, ) (H) = {g Q g h = h g, h H}, denotes the centralizer of the subset H Q in (Q, ).
25 Multiplication groups and generalized regular mappings Theorem Let (Q, ) be a quasigroup, let its isostrophe (Q, ) be a loop and α, β S Q. The following statements hold: 1. If x y = β(y)\α(x), then C SQ (RM(Q, )) = R ( ); 2. If x y = α(x)/β(y), then C SQ (RM(Q, )) = L ( ); 3. If x y = α(x)\β(y), then C SQ (LM(Q, )) = R ( ); 4. If x y = β(y)/α(x), then C SQ (LM(Q, )) = L ( ). Corollary 1. If (Q, ) is a middle Bol loop and (Q, ) is the corresponding right Bol loop, then C SQ (RM(Q, )) = R ( ) ; 2. If (Q, ) is a middle Bol loop and (Q, ) is the corresponding left Bol loop, then C SQ (RM(Q, )) = L ( ).
26 Multiplication groups and generalized regular mappings Theorem Let (Q, ) be a quasigroup, let its isostrophe (Q, ) be a loop and α, β S Q. The following statements hold: 1. If x y = β(y)\α(x), then C SQ (RM(Q, )) = R ( ); 2. If x y = α(x)/β(y), then C SQ (RM(Q, )) = L ( ); 3. If x y = α(x)\β(y), then C SQ (LM(Q, )) = R ( ); 4. If x y = β(y)/α(x), then C SQ (LM(Q, )) = L ( ). Corollary 1. If (Q, ) is a middle Bol loop and (Q, ) is the corresponding right Bol loop, then C SQ (RM(Q, )) = R ( ) ; 2. If (Q, ) is a middle Bol loop and (Q, ) is the corresponding left Bol loop, then C SQ (RM(Q, )) = L ( ).
27 Left, right and middle pseudoautomorphisms Let (Q, ) be an arbitrary loop and c Q. A mapping ϕ S Q is called a right (resp. left) pseudoautomorphism of (Q, ), with the companion c, if for every x, y Q the following equality holds: ϕ(x y) = (c ϕ(x)) ϕ(y) resp. ϕ(x y) = ϕ(x) (ϕ(y) c) Definition A mapping ϕ S Q is called a middle pseudoautomorphism of (Q, ), with the companion c, if for every x, y Q, the following equality holds: ϕ(x y) = [ϕ(x)/c 1 ] [c\ϕ(y)], where c 1 is the right inverse of c.
28 Left, right and middle pseudoautomorphisms Let (Q, ) be an arbitrary loop and c Q. A mapping ϕ S Q is called a right (resp. left) pseudoautomorphism of (Q, ), with the companion c, if for every x, y Q the following equality holds: ϕ(x y) = (c ϕ(x)) ϕ(y) resp. ϕ(x y) = ϕ(x) (ϕ(y) c) Definition A mapping ϕ S Q is called a middle pseudoautomorphism of (Q, ), with the companion c, if for every x, y Q, the following equality holds: ϕ(x y) = [ϕ(x)/c 1 ] [c\ϕ(y)], where c 1 is the right inverse of c.
29 On the group of middle pseudoautomorphisms of an arbitrary loop Proposition Let (Q, ) be an arbitrary loop with the unit e and let ϕ and ψ be middle pseudoautomorphisms of (Q, ), with the companions c and b, respectively. The following statements hold: 1. ϕ 1 is a middle pseudoautomorphism of (Q, ) with the companion ϕ 1 (c\e); 2. ϕψ is a middle pseudoautomorphism of (Q, ) with the companion R 1 c\e ϕr 1 b\e (e); 3. The set of middle pseudoautomorphisms of an arbitrary loop is a group.
30 Middle pseudoautomorphisms in right (left) Bol loops Denote by PS ( ) r (resp. PS ( ) l, PS ( ) m ) the group of all right (resp. left, middle) pseudoautomorphisms of the loop (Q, ). Remark Let (Q, ) be a right (left) Bol loop and c Q. A mapping ϕ S Q is a middle pseudoautomorphism of (Q, ), with the companion c, if and only if ϕ is a right (resp. left) pseudoautomorphism with the companion c (resp. c 1 ). So, PS ( ) m = PS ( ) r (resp. PS ( ) m = PS ( ) l ).
31 Middle pseudoautomorphisms in right (left) Bol loops Denote by PS ( ) r (resp. PS ( ) l, PS ( ) m ) the group of all right (resp. left, middle) pseudoautomorphisms of the loop (Q, ). Remark Let (Q, ) be a right (left) Bol loop and c Q. A mapping ϕ S Q is a middle pseudoautomorphism of (Q, ), with the companion c, if and only if ϕ is a right (resp. left) pseudoautomorphism with the companion c (resp. c 1 ). So, PS ( ) m = PS ( ) r (resp. PS ( ) m = PS ( ) l ).
32 Connections between groups of pseudoautomorphisms of left, right and middle Bol loops Proposition Let (Q, ) be a right Bol loop and let (Q, ) be the corresponding middle Bol loop. The following statements are true: 1. PS ( ) m = PS ( ) r = PS ( ) r ; 2. PS ( ) l = PS ( ) m ; 3. α PS ( ) r IαI PS ( ) l ; 4. α PS ( ) IαI PS ( ) 5. PS ( ) r 6. PS ( ) l l = PS ( ) l ; = PS ( ) r. r ;
33 Bibliography 1. Belousov V., Foundations of the theory of quasigroups and loops, Nauka, Moscow, 1967.(Russian) 2. Grecu I. and Syrbu P., Commutants of middle Bol loops. Quasigroups and Related Systems 22 (2014), Grecu I., Syrbu P. On some isostrophy invariants of Bol loops. Bulletin of the Transilvania University of Braşov, Series III: Math., Inf., Phys., Vol 5(54) 2012, Greer M., Kinyon M. Pseudoautomorphisms of Bruck loops and their generalizations. Commentationes Mathematicae Universitatis Carolinae 53.3 (2012): Kinyon M. K. and Phillips J. D., Commutants of Bol loops of odd order. Proc. Amer. Math. Soc. 132 (2004), Kinyon M. K., Phillips J. D., Vojtechovsky P. When is the commutant of a Bol loop a subloop. Trans. Amer. Math. Soc. 360 (2008), Nagy G.P. A class of simple proper Bol loops. Manuscripta Math., 127(2008), No. 1, 81 88,( arxiv:math/ v1[math.gr]). 8. Syrbu P. On middle Bol loops, ROMAI J.,no. 6, 2 (2010),
34 Thank you for your attention
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