Deciding active structural completeness

Stronkowski, Michal M.

doi:10.1007/s00153-019-00682-x

Cited by 2 publications

(2 citation statements)

References 24 publications

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“…also [64]. Computational aspects of these notions are explored in [17,46,69]. 4 Evidently, a quasi-equation is passive over an algebraic quasivariety K if and only if it is passive over the variety V(K).…”

Section: Definition 71mentioning

confidence: 99%

Singly generated quasivarieties and residuated structures

Moraschini

Raftery

Wannenburg

2020

Mathematical Logic Qtrly

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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.

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Section: Definition 71mentioning

confidence: 99%

Singly generated quasivarieties and residuated structures

Moraschini

Raftery

Wannenburg

2020

Mathematical Logic Qtrly

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“…(A complementary demand, now called 'active structural completeness' and analysed in [17], asks that condition (iii) of Theorem 6.1 should hold for all active quasi-equations; also see [60]. Computational aspects of [active] structural completeness are addressed in [16,43,64]. )…”

Section: Passive Structural Completenessmentioning

confidence: 99%

Singly generated quasivarieties and residuated structures

Moraschini

Raftery

Wannenburg

2019

Preprint

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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.

show abstract

Deciding active structural completeness

Cited by 2 publications

References 24 publications

Singly generated quasivarieties and residuated structures

Singly generated quasivarieties and residuated structures

Singly generated quasivarieties and residuated structures

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