Medial quandles are represented using a heterogeneous affine structure. As a consequence, we obtain numerous structural properties, including enumeration of isomorphism classes of medial quandles up to 13 elements.

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.

This paper is devoted to the semilattice ordered V-algebras of the form ( A, , +), where + is a join-semilattice operation and ( A, ) is an algebra from some given variety V. We characterize the free semilattice ordered algebras using the concept of extended power algebras. Next we apply the result to describe the lattice of subvarieties of the variety of semilattice ordered V-algebras in relation to the lattice of subvarieties of the variety V.

We find a syntactic characterization of the class of lattices embeddable into convexity lattices of posets which are trees. The characterization implies that this class forms a finitely based variety.

In [9] Etingof, Schedler and Soloviev introduced, for each non-degenerate involutive set-theoretical solution (X, σ, τ ) of the Yang-Baxter equation, the equivalence relation ∼ defined on the set X and they considered a new non-degenerate involutive induced retraction solution defined on the quotient set X ∼ . 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 ≈ on a birack, which is equal to ∼ in an involutive case. We present a complete algebraic proof that the relation ≈ 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.