A set-theoretic solution to the Pentagon Equation can be described as a pentagon algebra (S, · , *) such that (S, · ) is a semigroup and the operations · and * are related by two additional equations. This paper aims to investigate associative pentagon algebras in which (S, *) is also a semigroup. We introduce and describe two families of associative pentagon algebras which are strongly determined by the properties of the semigroup (S,*) . We present a complete characterization of such algebras using semigroup equations. We also provide constructions of such associative pentagon algebras and give several classes of examples.

We study non-degenerate set-theoretic solutions of the Yang–Baxter equation of multipermutation level 2 which are not 2-reductive. We describe an effective way of constructing such solutions using square-free 2-reductive solutions and two bijections. We present an algorithm how to obtain all such finite solutions, up to isomorphism. Using this algorithm, we enumerate all solutions of multipermutation level 2 up to size 6.

We present a general description of the notion of a retract of an algebra with a set of binary operations and describe an algorithm for finding the retract. We apply the result to fill a gap and give a correct definition of multipermutation degenerate set-theoretical solutions of the Yang-Baxter equation which has not been established so far.

Most of the set-theoretical solutions of the Yang-Baxter equation studied in the past years were non-degenerate multipermutation solutions. For degenerate solutions, a correct definition of multipermutation solutions has not been established so far. We fill here this gap providing a definition of multipermutation solutions that generalizes the one for non-degenerate solutions and we find an axiomatic description of this class by a set of equations that generalizes the equations describing non-degenerate multipermutation solutions. It turned out that the results do not need all the properties of solutions of the Yang-Baxter equation and therefore we prove them in a general universal-algebraic setting.

We present a complete characterization of all indecomposable non-degenerate, not necessarily involutive, solutions of the Yang-Baxter equation of multipermutation level 2. We show that every such solution is a homomorphic image of a special, “largest” solution called the universal one. On the other hand we prove that there is much simpler description. At first, on the product of a group Z_n^2 and an abelian group G, we construct some family of indecomposable non-degenerate solutions of the Yang-Baxter equation of multipermutation level 2. Next, applying Rosenbaum's theorem of subgroups of a semidirect product and isolating a triple: a subgroup of G, a subgroup of Z_n^2 and one group homomorphism, we obtain a full description of each epimorphism which gives the desired solutions. Such a construction provides a tool how to find (and possibly enumerate) all indecomposable non-degenerate solutions of multipermutation level 2. We also argue that the automorphism group of the discussed solutions is regular.

We study the diagonal mappings in non-involutive set-theoretic solutions of the Yang–Baxter equation. We show that, for non-degenerate solutions, they are commuting bijections. This gives the positive answer to the question: “Is every non-degenerate solution bijective?” of Cedó, Jespers and Verwimp. Additionally, we show that, for a subclass of solutions called k - permutational , only one-sided non-degeneracy suffices to prove that one of the diagonal mappings is invertible. We also present an equational characterization of multipermutation solutions and extend results of Rump, Gateva–Ivanova and Castelli, Mazzotta, Stefanelli about decomposability to non-involutive infinite case. In particular, we show that each, not necessarily involutive, square-free multipermutation solution of finite level and arbitrary cardinality, is always decomposable.

We study diagonal mappings in non-involutive set-theoretic solutions of the Yang-Baxter equation. We show that they are commuting bijections. We also give equational characterization of multipermutation solutions and extend results of Rump and Gateva-Ivanova about decomposability to non-involutive solutions. We show that each, not necessarily involutive, square-free multipermutation solution of arbitrary cardinality, is always decomposable.

We study 2-reductive non-involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation. We give a combinatorial construction of any such solution of any (even infinite) size. We also prove that solutions associated to a skew left brace are 2-reductive if and only if the skew left brace is nilpotent of class 2. Moreover, all such skew left braces are actually bi-skew left braces. We focus on these structures and we give several equivalent properties characterizing solutions associated to bi-skew left braces.

We give a complete characterization of all indecomposable involutive solutions of the Yang-Baxter equation of multipermutation level 2. In the first step we present a construction of some family of such solutions and in the second step we prove that every indecomposable involutive solution of the Yang-Baxter equation with multipermutation level 2 is a homomorphic image of a solution previously constructed. Analyzing this epimorphism, we are able to obtain all such solutions up to isomorphism and enumerate these of small sizes.

We study indecomposable involutive set-theoretic solutions of the Yang-Baxter equation with cyclic permutation groups (cocyclic solutions). In particular, we show that there is no one-to-one correspondence between indecomposable cocyclic solutions and cocyclic braces which contradicts recent results in \cite{Rump21}.

We study indecomposable involutive set-theoretic solutions of the Yang-Baxter equation with cyclic permutation groups (cocyclic solutions). In particular, we show that there is no one-to-one correspondence between indecomposable cocyclic solutions and cocyclic braces which contradicts recent results in [19].

We investigate a class of non-involutive solutions of the Yang–Baxter equation which generalize derived (self-distributive) solutions. In particular, we study generalized multipermutation solutions in this class. We show that the Yang–Baxter (permutation) groups of such solutions are nilpotent. We formulate the results in the language of biracks which allows us to apply universal algebra tools.

We study involutive set-theoretic solutions of the Yang-Baxter equation of multipermutation level 2. These solutions happen to fall into two classes – distributive ones and non-distributive ones. The distributive ones can be effectively constructed using a set of abelian groups and a matrix of constants. Using this construction, we enumerate all distributive involutive solutions up to size 14. The non-distributive solutions can be also easily constructed, using a distributive solution and a permutation.

We continue our studies on semilattice ordered algebras. This time we accept constants in the type of algebras. We investigate identities satisfied by such algebras and describe the free objects in varieties of semilattice ordered algebras with constants.

We characterize the para-associative ternary quasi-groups (flocks) applicable to knot theory, and show which of these structures are isomorphic. We enumerate them up to order 64. We note that the operation used in knot-theoretic flocks has its non-associative version in extra loops. We use a group action on the set of flock colorings to improve the cocycle invariant associated with the knot-theoretic flock (co)homology.

In [9] Etingof, Schedler and Soloviev introduced, for each non-degenerate involutive set-theoretical solution (X, sigma, tau) of the Yang-Baxter equation, the equivalence relation similar to defined on the set X and they considered a new non-degenerate involutive induced retraction solution defined on the quotient set X-similar to. It is well known that translating set-theoretical non-degenerate solutions of the Yang-Baxter equation into the universal algebra language we obtain an algebra called a birack. In the paper we introduce the generalized retraction relation approximate to on a birack, which is equal to similar to in an involutive case. We present a complete algebraic proof that the relation approximate to is a congruence of the birack. Thus we show that the retraction of a set-theoretical non-degenerate solution is well defined not only in the involutive case but also in the case of all non-involutive solutions. (C) 2018 Elsevier B.V. All rights reserved.

We describe various properties and give several characterizations of ternary groups satisfying two axioms derived from the third Reidemeister move in knot theory. Using special attributes of such ternary groups, such as semi-commutativity, we construct a ternary invariant of curves immersed in compact surfaces, considered up to flat Reidemeister moves.

A quandle will be called quasi-affine, if it embeds into an affine quandle. Our main result is a characterization of quasi-affine quandles, by group-theoretic properties of their displacement group, by a universal algebraic condition coming from the commutator theory, and by an explicit construction over abelian groups. As a consequence, we obtain efficient algorithms for recognizing affine and quasi-affine quandles, and we enumerate small quasi-affine quandles. We also prove that the “abelian implies quasi-affine” problem of universal algebra has affirmative answer for the class of quandles.

We describe all subdirectly irreducible medial quandles. We show that they fall within one of four disjoint classes. In particular, in the finite case they are either connected (and therefore Alexander quandles) or reductive. Moreover, we provide a representation of all non-connected subdirectly irreducible medial quandles.

This paper gives the construction of free medial quandles as well as free n-symmetric medial quandles and free m-reductive medial quandles.