Varieties of De Morgan monoids: Minimality and irreducible algebras

The Review of Symbolic Logic

Wannenburg

2019

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The variety DMM of De Morgan monoids has just four minimal subvarieties. The join-irreducible covers of these atoms in the subvariety lattice of DMM are investigated. One of the two atoms consisting of idempotent algebras has no such cover; the other has just one. The remaining two atoms lack nontrivial idempotent members. They are generated, respectively, by 4-element De Morgan monoids C4 and D4, where C4 is the only nontrivial 0-generated algebra onto which finitely subdirectly irreducible De Morgan monoids may be mapped by noninjective homomorphisms. The homomorphic preimages of C4 within DMM (together with the trivial De Morgan monoids) constitute a proper quasivariety, which is shown to have a largest subvariety U. The covers of the variety (C4) within U are revealed here. There are just ten of them (all finitely generated). In exactly six of these ten varieties, all nontrivial members have C4 as a retract. In the varietal join of those six classes, every subquasivariety is a variety—in fact, every finite subdirectly irreducible algebra is projective. Beyond U, all covers of (C4) [or of (D4)] within DMM are discriminator varieties. Of these, we identify infinitely many that are finitely generated, and some that are not. We also prove that there are just 68 minimal quasivarieties of De Morgan monoids.

“…it follows from Lemma 2.1(i) that A satisfies the same laws. 3 Then A satisfies (23) and (24), by (12), because e 1. Thus, A ∈ U.…”

Section: (I) Every Quasivariety Of Odd Sugihara Monoids Is a Varietymentioning

confidence: 97%

Section: (I) Every Quasivariety Of Odd Sugihara Monoids Is a Varietymentioning

confidence: 99%

“…M is the class of all algebras in U satisfying e f . In particular, M satisfies(22),(23) and(24).COROLLARY 4.15. Every rigorously compact algebra in N belongs to M. Proof.…”

mentioning

confidence: 97%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Varieties of De Morgan Monoids: Covers of Atoms

The Review of Symbolic Logic

Wannenburg

2019

Self Cite

Singly generated quasivarieties and residuated structures

Mathematical Logic Qtrly

Wannenburg

2020

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.

The algebraic significance of weak excluded middle laws

Lávička

Mathematical Logic Qtrly

2022

For (finitary) deductive systems, we formulate a signature‐independent abstraction of the weak excluded middle law (WEML), which strengthens the existing general notion of an inconsistency lemma (IL). Of special interest is the case where a quasivariety K$\mathsf {K}$ algebraizes a deductive system ⊢. We prove that, in this case, if ⊢ has a WEML (in the general sense) then every relatively subdirectly irreducible member of K$\mathsf {K}$ has a greatest proper K$\mathsf {K}$‐congruence; the converse holds if ⊢ has an inconsistency lemma. The result extends, in a suitable form, to all protoalgebraic logics. A super‐intuitionistic logic possesses a WEML iff it extends KC$\mathsf {KC}$. We characterize the IL and the WEML for normal modal logics and for relevance logics. A normal extension of sans-serifSsans-serif4$\mathsf {S4}$ has a global consequence relation with a WEML iff it extends sans-serifSsans-serif4.sans-serif2$\mathsf {S4.2}$, while every axiomatic extension of sans-serifRsans-serift$\mathsf {R^t}$ with an IL has a WEML.