The concept of pseudosquare in a general quadratical quasigroup is introduced and connections to some other geometrical concepts are studied. The geometrical presentations of some proved statements are given in the quadratical quasigroup $ℂ\left(\frac{1+i}{2}\right)$.

Slim groupoids are groupoids satisfying $x\left(yz\right)x̄z$. We find all simple slim groupoids and all minimal varieties of slim groupoids. Every slim groupoid can be embedded into a subdirectly irreducible slim groupoid. The variety of slim groupoids has the finite embeddability property, so that the word problem is solvable. We introduce the notion of a strongly nonfinitely based slim groupoid (such groupoids are inherently nonfinitely based) and find all strongly nonfinitely based slim groupoids with at most four...

G. Grätzer and A. Kisielewicz devoted one section of their survey paper concerning $p_n$-sequences and free spectra of algebras to the topic “Small idempotent clones” (see Section 6 of [18]). Many authors, e.g., [8], [14, 15], [22], [25] and [29, 30] were interested in $p_n$-sequences of idempotent algebras with small rates of growth. In this paper we continue this topic and characterize all idempotent groupoids $(G,·)$ with $p_2(G,·)\le 2$ (see Section 7). Such groupoids appear in many papers see, e.g. [1], [4], [21], [26,...

The remarkable development of the theory of smooth quasigroups is surveyed.

We shall show that there exist sofic groups which are not locally embeddable into finite Moufang loops. These groups serve as counterexamples to a problem and two conjectures formulated in the paper by M. Vodička, P. Zlatoš (2019).

The authors prove that a local n-quasigroup defined by the equation $x_{n+1}=F(x₁,...,xₙ)=(f₁\left(x₁\right)+...+fₙ\left(xₙ\right))/(x₁+...+xₙ)$, where $f_i\left(x_i\right)$, i,j = 1,...,n, are arbitrary functions, is irreducible if and only if any two functions $f_i\left(x_i\right)$ and $f_j\left(x_j\right)$, i ≠ j, are not both linear homogeneous, or these functions are linear homogeneous but $f_i\left(x_i\right)/x_i\ne f_j\left(x_j\right)/x_j$. This gives a solution of Belousov’s problem to construct examples of irreducible n-quasigroups for any n ≥ 3.

We are investigating quasigroup functional equation classification up to parastrophic equivalence [Sokhatsky F.M.: On classification of functional equations on quasigroups, Ukrainian Math. J. 56 (2004), no. 4, 1259–1266 (in Ukrainian)]. If functional equations are parastrophically equivalent, then their functional variables can be renamed in such a way that the obtained equations are equivalent, i.e., their solution sets are equal. There exist five classes of generalized distributive-like quasigroup...

This paper presents some manner of characterization of Boolean rings. These algebraic systems one can also characterize by means of some distributivities satisfied in GBbi-QRs.

In this paper we show that there exists an infinite family of pairwise non-isomorphic entropic quasigroups with quasi-identity which are directly indecomposable and they are two-generated.

Irregular (quasi)varieties of groupoids are (quasi)varieties that do not contain semilattices. The regularization of a (strongly) irregular variety $𝒱$ of groupoids is the smallest variety containing $𝒱$ and the variety $𝒮$ of semilattices. Its quasiregularization is the smallest quasivariety containing $𝒱$ and $𝒮$. In an earlier paper the authors described the lattice of quasivarieties of cancellative commutative binary modes, i.e. idempotent commutative and entropic (or medial) groupoids. They are all irregular...

In this paper we make some remarks on simple Bol loops which were motivated by questions at the LOOPS'07 conference. We also list some open problems on simple loops.