Protomodular categories were introduced by the first author more than ten years ago. We show that a variety V of universal algebras is protomodular if and only if it has 0-ary terms e1, . . . , en, binary terms t1, . . . , tn, and (n+1)-ary term t satisfying the identities t(x, t1(x, y), . . . , tn(x, y)) = y and ti(x, x) = ei for each i = 1, . . . , n.

Codescent morphisms are described in regular categories which satisfy the so-called strong amalgamation property. Among varieties of universal algebras possessing this property are, as is known, categories of groups, not necessarily associative rings, M -sets (for a monoid M), Lie algebras (over a field), quasi-groups, commutative quasi-groups, Steiner quasi-groups, medial quasigroups, semilatt...

A category is said to be alg-universal, if every category of universal algebras can be fully embedded into it. We prove here that the category of varieties and interpretations, or in other words, the category of abstract clones and clone homomorphisms, is alg-universal.

We re-examine some topics in representation theory of Lie algebras and Springer theory in a more general context, viewing the universal enveloping algebra as an example of the section ring of a quantization of a conical symplectic resolution. While some modification from this classical context is necessary, many familiar features survive. These include a version of the Beilinson-Bernstein local...

Following work of Pa lasińska and Pigozzi on partially ordered varieties and quasi-varieties of universal algebras, the author recently introduced partially ordered systems (posystems) and partially ordered functors (pofunctors) to cover the case of the algebraic systems arising in categorical abstract algebraic logic. Analogs of the ordered homomorphism theorems of universal algebra were shown...

We study closedness properties of internal relations in finitely complete categories, which leads to developing a unified approach to: Mal’tsev categories, in the sense of A.Carboni, J. Lambek and M.C.Pedicchio, that generalize Mal’tsev varieties of universal algebras; unital categories, in the sense of D.Bourn, that generalize pointed Jónsson-Tarski varieties; and subtractive categories, intro...

We study exponentiability of homomorphisms in varieties of universal algebras close to classical ones. After describing an “almost folklore” general result, we present a purely algebraic proof of “étale implies exponentiable”, alternative to the topologically motivated proof given in one of our previous papers, in a different context. We prove that only isomorphisms are exponentiable homomorphi...

Trees are among the most fundamental and ubiquitous structures in mathematics. Tree languages and automata on trees have been studied extensively since the 1960s from both a purely mathematical and application point of view. When trees are defined as terms, universal algebra becomes directly applicable to tree automata and, on the other hand, the theory of tree automata suggests new notions and...