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...

In any recursive algebraic language, I find an interval of the lattice of equational theories, every element of which has finitely many covers. With every finite set of equations of this language, an equational theory of this interval is associated, which is decidable with decidable covers that can be algorithmically found. If the language is finite, both this theory and its covers are finitely based. Also, for every finite language and for every natural number n, I construct a finitely based decidable...

On the lattice $\mathcal{L}\left(\mathcal{C}\mathcal{R}\right)$ of varieties of completely regular semigroups considered as algebras with the binary multiplication and unary inversion within maximal subgroups, we study the relations $K_l$, $K$, $K_r$, $T_l$, $T$, $T_r$, $C$ and $L$. Here $K$ is the kernel relation, $T$ is the trace relation, $T_l$ and $T_r$ are the left and the right trace relations, respectively, $K_p=K\cap T_p$ for $p\in \left\{l,r\right\}$, $C$ is the core relation and $L$ is the local relation. We give an alternative definition for each of these relations $P$ of the form $$\mathcal{U}P\mathcal{V}\iff \mathcal{U}\cap \tilde{P}=\mathcal{V}\cap \tilde{P}(\mathcal{U},\mathcal{V}\in \mathcal{L}(\mathcal{C}\mathcal{R}\left)\right),$$ for some subclasses $\tilde{P}$ of $\mathcal{C}\mathcal{R}$....

Graph algebras establish a connection between directed graphs without multiple edges and special universal algebras of type (2,0). We say that a graph G satisfies a term equation s ≈ t if the corresponding graph algebra $\underline{A\left(G\right)}$ satisfies s ≈ t. A class of graph algebras V is called a graph variety if $V=Mo{d}_g\Sigma$ where Σ is a subset of T(X) × T(X). A graph variety $V^{\text{'}}=Mo{d}_g{\Sigma }^{\text{'}}$ is called a biregular leftmost graph variety if Σ’ is a set of biregular leftmost term equations. A term equation s ≈ t is called an identity in a variety...

J. Płonka in [12] noted that one could expect that the regularization $ℛ\left(K\right)$ of a variety $K$ of unary algebras is a subdirect product of $K$ and the variety $D$ of all discrete algebras (unary semilattices), but is not the case. The purpose of this note is to show that his expectation is fulfilled for those and only those irregular varieties $K$ which are contained in the generalized variety $TDir$ of the so-called trap-directable algebras.