Structural Completeness in Relevance Logics

Mathematical Logic Qtrly

Raftery

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.

“…Here, A is not

{bold-italicF}_{sans-serifRA} false(ℵ_{0} false)

, as

sans-serifRA

is not (even passively) structurally complete. In fact,

sans-serifRA

has no nontrivial PSC subvariety, other than

V (2)

[62, Theorem 6]. This contrasts with the fact that

sans-serifDMM

lacks the JEP (by Proposition 5.5(i), as it has non‐isomorphic 0‐generated nontrivial members).…”

Section: De Morgan Monoids: a Case Studymentioning

confidence: 99%

“…Here, A is not F RA (ℵ 0 ), as RA is not (even passively) structurally complete. In fact, RA has no nontrivial PSC subvariety, other than V(2) [62,Theorem 6].…”

Section: Theorem 84mentioning

confidence: 99%

Singly generated quasivarieties and residuated structures

Mathematical Logic Qtrly

Raftery

Wannenburg

2020

“…of quasi-equations admissible in the variety of Heyting algebras) and of some prominent modal systems have been provided in [29,31]. Unfortunately, the picture of structural completeness tends to change dramatically when the set of connectives is altered (see, for instance, [53]). Accordingly, we should not expect, at least in principle, that results on admissibility and structural completeness valid in modal algebras could be extended directly to positive modal algebras.…”

Section: K Is Hsc If and Only If Every Subquasi-variety Of K Is A Varmentioning

confidence: 99%

Varieties of Positive Modal Algebras and Structural Completeness

The Review of Symbolic Logic

2019

Positive modal algebras are the ∧, ∨, 3, , 0, 1 -subreducts of modal algebras. We prove that the variety of positive S4-algebras is not locally finite. On the other hand, the free one-generated positive S4-algebra is shown to be finite. Moreover, we describe the bottom part of the lattice of varieties of positive S4-algebras. Building on this, we characterize (passively, hereditarily) structurally complete varieties of positive K4-algebras. 1 arXiv:1908.01659v1 [math.LO] 1 Aug 2019Definition 3.2. A K + -space is a structure X, , R, τ where X, , τ is a Priestley space and R is a binary relation on X such that:The clopen upsets of τ are closed under the operations R and Q R . 3. The set {y ∈ X : x, y ∈ R} is topologically closed for every x ∈ X. Definition 3.3. Let X

“…This A is not F RA (ℵ 0 ), as RA is not (even passively) structurally complete. In fact, RA has no nontrivial PSC subvariety, other than V(2) [58,Thm. 6].…”

Section: 7])mentioning

confidence: 99%

Singly generated quasivarieties and residuated structures

Raftery

Wannenburg

2019

Preprint

Self Cite

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 ℵ0-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.