Singly generated quasivarieties and residuated structures

Abstract

A quasivariety sans-serifK of algebras has the joint embedding property (JEP) if and only if it is generated by a single algebra A. It is structurally complete if and only if the free ℵ0‐generated algebra in sans-serifK can serve as A. A consequence of this demand, called ‘passive structural completeness’ (PSC), is that the nontrivial members of sans-serifK all satisfy the same existential positive sentences. We prove that if sans-serifK 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 sans-serifK is relatively semisimple then it is generated by one sans-serifK‐simple algebra. We also prove that a quasivariety of finite type, with a finite nontrivial member, is PSC if and only if 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.

Structural completeness in many-valued logics with rational constants

“…If an HSC logic is algebraized by a quasivariety K, then the lattice of extensions of and Q(K) are distributive. Proposition 3.4 ([63,Thm. 4.3 & Rmk.…”

“…Admissibility in extensions of Ł was investigated in [35,36]. 2 Finally, works addressing variants of structural completeness, such as active and passive rules also studied in this paper, include [24,37,62,63,71,80].…”

The algebraic significance of weak excluded middle laws

“…Some characterizations of the Kollár subvarieties of $$\mathsf {DMM}$$ will be provided. Where known structural features of De Morgan monoids are mentioned below without citation, their sources are given in the recent papers [31–33]. Definition A De Morgan monoid is an algebra $$\bm {A}=\langle A;\mathbin {\bm{\cdot }},\wedge ,\vee ,\lnot ,e\rangle$$ comprising a distributive lattice $$\langle A;\wedge ,\vee \rangle$$, a commutative monoid $$\langle A;\mathbin {\bm{\cdot }},e\rangle$$ that is square‐increasing (i.e., $$\bm {A}$$ satisfies $$x\leqslant x^2\coloneqq x\mathbin {\bm{\cdot }}x$$), and a function $$\lnot : A\mathrel {\longrightarrow }A$$, called an involution , such that $$\bm {A}$$ satisfies $$\lnot \lnot x= x$$ and $$\begin{equation*} x\mathbin {\bm{\cdot }}y\leqslant z\;\Longleftrightarrow \;x\mathbin {\bm{\cdot }}\lnot z\leqslant \lnot y.$$…”

“…A quasivariety $$\mathsf {K}$$ of De Morgan monoids is a Kollár quasivariety iff $$\bm {S}_3\notin \mathsf {K}$$ [33, Theorem 8.4(iii)]. Many such non‐semisimple varieties are exhibited in [32].…”

Varieties of Positive Modal Algebras and Structural Completeness

“…For instance, if a quasi-variety K is HSC, then its lattice of subquasi-varieties happens to be distributive [26,27]. Moreover, if K is PSC, then all its subquasi-varieties have the joint embedding property [44] and, therefore, are generated as quasi-varieties by a single algebra [39]. On the other hand, variants of structural completeness are interesting also from a purely logical perspective.…”