Units in quasigroups with Bol-Moufang type identities Natalia N. - PowerPoint PPT Presentation

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal Units in quasigroups with Bol-Moufang type identities Natalia N. Didurik ∗ and Victor A. Shcherbacov ⋆ Dimitrie Cantemir State University ∗ and Institute of Mathematics and Computer Science ⋆ Budapest, July 11, 2019 Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University ∗ and Institute of Mathematics and Computer Science ⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal Objects Groupoids and quasigroups Groupoids Definition A binary groupoid ( G , A ) is a non-empty set G together with binary operation A . We give main definition of a quasigroup: Definition Binary groupoid ( Q , ◦ ) is called a quasigroup if for any ordered pair ( a , b ) ∈ Q 2 there exist unique solutions x , y ∈ Q to the equations x ◦ a = b and a ◦ y = b [2]. Often this definition is called existential . In fact this definition was given by Ruth Moufang. Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University ∗ and Institute of Mathematics and Computer Science ⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal Objects Units Definitions Definition ◮ An element f ∈ Q is a left unit of ( Q ; · ) iff f · x = x for all x ∈ Q . ◮ An element e ∈ Q is a right unit of ( Q ; · ) iff x · e = x for all x ∈ Q . ◮ An element s ∈ Q is a middle unit of ( Q ; · ) iff x · x = s for all x ∈ Q . ◮ A quasigroup which has left unit is called here left loop. ◮ A quasigroup which has right unit is called here right loop. ◮ A quasigroup ( Q , · ) with an identity element e ∈ Q is called a loop . Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University ∗ and Institute of Mathematics and Computer Science ⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal Objects Introduction An algebra with three binary operations ( Q , · , /, \ ) that satisfies the following identities: x · ( x \ y ) = y , (1) ( y / x ) · x = y , (2) x \ ( x · y ) = y , (3) ( y · x ) / x = y , (4) is called a quasigroup ( Q , · , /, \ ) [2]. Basic facts about quasigroups and loops can be found in [2]. Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University ∗ and Institute of Mathematics and Computer Science ⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal Objects Definition Let ( G , · ) be a groupoid and let a be a fixed element in G . Translation maps L a (left) and R a (right) are defined by L a x = a · x , R a x = x · a for all x ∈ G . For quasigroups it is possible to define a third kind of translation, namely, middle translations. If P a is a middle translation of a quasigroup ( Q , · ), then x · P a x = a for all x ∈ Q [3]. Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University ∗ and Institute of Mathematics and Computer Science ⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal Objects Belousov problems Belousov’s Problem #18. How to recognize identities which force quasigroups satisfying them to be loops [2]? In the first place we note the result of J.D.H. Smith [26, Proposition 1.3]. A nonempty quasigroup ( Q , · , /, \ ) is a loop if and only if it satisfies the “slightly associative identity” x ( y / y ) · z = x · ( y / y ) z . (5) You understand that previous theorem does not solve Belousov problem, but this is an important criterion. Notice, identity (5) is the generalized Bol-Moufang type identity in the language of this presentation, Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University ∗ and Institute of Mathematics and Computer Science ⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal Objects Generalized Belousov problems Generalized Belousov’s Problem #18. How to recognize identities which force quasigroups satisfying them to have left, right, middle unit [17]? In the article [17] we try to research generalized Belousov’s Problem #18 for quasigroups with identities of two variables and one fixed element. For example, we research which units has quasigroup ( Q , · ) with the following identity and with fixed element a : ax · y = a · xy . (6) Here letters x , y denote free variables, and the letter a denotes a fixed element of the set Q . Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University ∗ and Institute of Mathematics and Computer Science ⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal Objects Belousov problems G.B. Belyavskaya [5, 6] studied identities with permutations. An identity with permutations can be defined as an equality between two terms that contain variables and symbols of quasigroup operations · , /, \ and permutations of the set Q incorporated in this equality. We can re-write identity (6) in the following form: L a x · y = L a ( xy ) . (7) So identity (7) is an identity with permutation. Implicitly this concept is well known in quasigroup theory. For example, identities with permutations are used in the definition of an IP -loop [R. Moufang, Math. Ann. 110 (1935), no. 1, 416–430;]. Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University ∗ and Institute of Mathematics and Computer Science ⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal Objects Belousov problems It is clear that we can see on the equality (6) as on an autotopy ( L a , ε, L a ) of a quasigroup ( Q , · ). We can say that equality (6) defines the left nuclear element a . Moreover, we can rewrite equality (6) in the form ax · y = a · ( x ◦ y ) . (8) In this case we obtain right derivative groupoid (operation) ( Q , ◦ ) of quasigroup ( Q , · ) [2, 24, 17]. In the article of Alexandar Krapez [16] generalized Belousov problem is researched for groupoids using generalized derivatives and functional equations. Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University ∗ and Institute of Mathematics and Computer Science ⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal Objects Theorem Autotopy f e s Autotopy f e s ( L a , L a , ε ) + − − ( L a , ε, L a ) + − − ( L a , L − 1 ( L a , ε, L − 1 a , ε ) + − − a ) + − − ( L a , R a , ε ) − − − ( L a , ε, R a ) + − − ( L a , R − 1 ( L a , ε, R − 1 a , ε ) − − − a ) + − − ( L a , P a , ε ) + − − ( L a , ε, P a ) − − − ( L a , P − 1 ( L a , ε, P − 1 a , ε ) + − − a ) − − − Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University ∗ and Institute of Mathematics and Computer Science ⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal Definitions An identity based on a single binary operation is of Bol-Moufang type if “both sides consist of the same three different letters taken in the same order but one of them occurs twice on each side”[9]. We use list of 60 Bol-Moufang type identities given in [11]. There exist other (“more general”) definitions of Bol-Moufang type identities and, therefore, other lists and classifications of such identities [1, 8]. Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University ∗ and Institute of Mathematics and Computer Science ⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal Definitions We recall, (12)-parastroph of groupoid ( G , · ) is a groupoid ( G , ∗ ) such that the operation “ ∗ ” is obtained by the following rule: x ∗ y = y · x . (9) It is clear that for any groupoid ( G , · ) there exists its (12)-parastroph groupoid ( G , ∗ ). Suppose that an identity F is true in groupoid ( G , · ). Then we can obtain (12)-parastrophic identity ( F ∗ ) of the identity F replacing the operation “ · ” with the operation “ ∗ ” and changing the order of variables following rule (9). Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University ∗ and Institute of Mathematics and Computer Science ⋆