Quasi-varieties of first-order structures were studied by N. Weaver [7] to generalize varieties of algebras; he also established some Malcev like conditions for these classes of structures. Following this line we extend some results of functional completeness of algebras to firstorder structures. Specifically, we formulate and characterize a notion of quasiprimality for first-order structures.

It is shown how Lawvere’s one-to-one translation between Birkhoff’s description of varieties and the categorical one (see [6]) turns Hu’s theorem on varieties generated by a primal algebra (see [4], [5]) into a simple reformulation of the classical representation theorem of finite Boolean algebras as powerset algebras.

We characterise quasivarieties and varieties of ordered algebras categorically in terms of regularity, exactness and the existence of a suitable generator. The notions of regularity and exactness need to be understood in the sense of category theory enriched over posets. We also prove that finitary varieties of ordered algebras are cocompletions of their theories under sifted colimits (again, i...

We consider several aspects of Wilke’s [T. Wilke, An algebraic characterization of frontier testable tree languages, Theoret. Comput. Sci. 154 (1996) 85–106] tree algebra formalism for representing binary labelled trees and compare it with approaches that represent trees as terms in the traditional way. A convergent term rewriting system yields normal form representations of binary trees and co...

It is known that exactly eight varieties of Heyting algebras have a modelcompletion, but no concrete axiomatisation of these model-completions were known by now except for the trivial variety (reduced to the one-point algebra) and the variety of Boolean algebras. For each of the six remaining varieties we introduce two axioms and show that 1) these axioms are satisfied by all the algebras in th...

In this paper we overview recent results about the lattice of subvarieties of the variety BL of BL-algebras and the equational definition of some families of them.

1. Congruence lattices. G. Birkhoff and O. Frink noted that the congruence lattice Con04) of an algebra A (with operations of finite rank) is algebraic or compactly generated. The celebrated Grätzer-Schmidt theorem states that, conversely, every algebraic lattice is isomorphic to Con(A) for some algebra A. The importance of this is obvious, for it shows that unless something more is known about...

We develop a new centrality concept and apply it to solve certain outstanding problems about nite algebras. In particular, we describe all nite algebras of nite complexity and all nite strongly abelian algebras which generate residually small varieties.

This is an overview article on finite-dimensional algebras and quivers, written for the Encyclopedia of Mathematical Physics. We cover path algebras, Ringel-Hall algebras and the quiver varieties of Lusztig and Nakajima.