Non-distributive Variety Research Articles

The Jónsson distributivity spectrum

Suppose throughout that $\mathcal V$ is a congruence distributive variety. If $m \geq 1$, let $ J _{ \mathcal V} (m) $ be the smallest natural number $k$ such that the congruence identity $\alpha ( \beta \circ \gamma \circ \beta \dots ) \subseteq \alpha \beta \circ \alpha \gamma \circ \alpha \beta \circ \dots $ holds in $\mathcal V$, with $m$ occurrences of $ \circ$ on the left and $k$ occurrences of $\circ$ on the right. We show that if $ J _{ \mathcal V} (m) =k$, then $ J _{ \mathcal V} (m \ell ) \leq k \ell $, for every natural number $\ell$. The key to the proof is an identity which, through a variety, is equivalent to the above congruence identity, but involves also reflexive and admissible relations. If $ J _{ \mathcal V} (1)=2 $, that is, $\mathcal V$ is $3$-distributive, then $ J _{ \mathcal V} (m) \leq m $, for every $m \geq 3$ (actually, a more general result is presented which holds even in nondistributive varieties). If $\mathcal V$ is $m$-modular, that is, congruence modularity of $\mathcal V$ is witnessed by $m+1$ Day terms, then $ J _{ \mathcal V} (2) \leq J _{ \mathcal V} (1) + 2m^2-2m -1 $. Various problems are stated at various places.

On the Structure of Equationally Complete Varieties. II

Each member $\mathcal {V}$ of a large family of nonassociative or, when applicable, nondistributive varieties has the following universal property: Every variety $\mathcal {K}$ that satisfies certain very weak versions of the amalgamation and joint embedding properties is isomorphic, as a category, to a coreflective subcategory of some equationally complete subvariety of $\mathcal {V}$. Moreover, the functor which serves to establish the isomorphism preserves injections. As a corollary one obtains the existence of equationally complete subvarieties of $\mathcal {V}$ that fail to have the amalgamation property and fail to be residually small. The family of varieties universal in the above sense includes commutative groupoids, bisemigroups (i.e., algebras with two independent associative operations), and quasi-groups.

On the structure of equationally complete varieties. II

Each member $\mathcal {V}$ of a large family of nonassociative or, when applicable, nondistributive varieties has the following universal property: Every variety $\mathcal {K}$ that satisfies certain very weak versions of the amalgamation and joint embedding properties is isomorphic, as a category, to a coreflective subcategory of some equationally complete subvariety of $\mathcal {V}$. Moreover, the functor which serves to establish the isomorphism preserves injections. As a corollary one obtains the existence of equationally complete subvarieties of $\mathcal {V}$ that fail to have the amalgamation property and fail to be residually small. The family of varieties universal in the above sense includes commutative groupoids, bisemigroups (i.e., algebras with two independent associative operations), and quasi-groups.