Singly generated quasivarieties and residuated structures

A quasivariety K of algebras has the joint embedding property (JEP) iff it is generated by a single algebra A. It is structurally complete iff the free countably generated algebra in K can serve as A. A consequence of this demand, called "passive structural completeness" (PSC), is that the nontrivial members of K all satisfy the same existential positive sentences. We prove that if K is PSC then it still has the JEP, and if it has the JEP and its nontrivial members lack trivial subalgebras, then its relatively simple members all belong to the universal class generated by one of them. Under these conditions, if K is relatively semisimple then it is generated by one K-simple algebra. It is a minimal quasivariety if, moreover, it is PSC but fails to unify some finite set of equations. We also prove that a quasivariety of finite type, with a finite nontrivial member, is PSC iff its nontrivial members have a common retract. The theory is then applied to the variety of De Morgan monoids, where we isolate the sub(quasi)varieties that are PSC and those that have the JEP, while throwing fresh light on those that are structurally complete. The results illuminate the extension lattices of intuitionistic and relevance logics

A note on drastic product logic

The drastic product $*_D$ is known to be the smallest $t$-norm, since $x *_D y = 0$ whenever $x, y < 1$. This $t$-norm is not left-continuous, and hence it does not admit a residuum. So, there are no drastic product $t$-norm based many-valued logics, in the sense of [EG01]. However, if we renounce standard completeness, we can study the logic whose semantics is provided by those MTL chains whose monoidal operation is the drastic product. This logic is called ${\rm S}_{3}{\rm MTL}$ in [NOG06]. In this note we justify the study of this logic, which we rechristen DP (for drastic product), by means of some interesting properties relating DP and its algebraic semantics to a weakened law of excluded middle, to the $\Delta$ projection operator and to discriminator varieties. We shall show that the category of finite DP-algebras is dually equivalent to a category whose objects are multisets of finite chains. This duality allows us to classify all axiomatic extensions of DP, and to compute the free finitely generated DP-algebras.Comment: 11 pages, 3 figure

The General Apple Property and Boolean terms in Integral Bounded Residuated Lattice-ordered Commutative Monoids

In this paper we give equational presentations of the varieties of {\em integral bounded residuated lattice-ordered commutative monoids} (bounded residuated lattices for short) satisfying the \emph{General Apple Property} (GAP), that is, varieties in which all of its directly indecomposable members are local. This characterization is given by means of Boolean terms: \emph{A variety $\mathsf{V}$ of \brl s has GAP iff there is an unary term $b(x)$ such that $\mathsf{V}$ satisfies the equations $b(x)\lor\neg b(x)\approx \top$ and $(x^k\to b(x))\cdot(b(x)\to k.x)\approx \top$, for some $k>0$}. Using this characterization, we show that for any variety $\mathsf{V}$ of bounded residuated lattice satisfying GAP there is $k>0$ such that the equation $k.x\lor k.\neg x\approx \top$ holds in $\mathsf{V}$, that is, $\mathsf{V} \subseteq \mathsf{WL_\mathsf{k}}$. As a consequence we improve Theorem 5.7 of \cite{CT12}, showing in theorem that a\emph{ variety of \brls\ has Boolean retraction term if and only if there is $k>0$ such that it satisfies the equation $k.x^k\lor k.(\neg k)^n\approx\top$.} We also see that in Bounded residuated lattices GAP is equivalent to Boolean lifting property (BLP) and so, it is equivalent to quasi-local property (in the sense of \cite{GLM12}). Finally, we prove that a variety of \brl s has GAP and its semisimple members form a variety if and only if there exists an unary term which is simultaneously Boolean and radical for this variety.Comment: 25 pages, 1 figure, 2 table