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

Models of Relevant Arithmetic

It is well known that the relevant arithmetic R# admits finite models whose domains are the integers modulo n rather than the expected natural numbers. Less well appreciated is the fact that the logic of these models is much more subtle than that of the three-valued structure in which they are usually presented. In this paper we consider the DeMorgan monoids in which R# can be modelled, deriving a fairly complete account of those modelling the stronger arithmetic RM# modulo n and a partial account for the case of R# modulo a prime number. The more general case in which the modulus is arbitrary is shown to lead to infinite propositional structures even with the additional constraint that '0=1' implies everything

Models of Relevant Arithmetic

It is well known that the relevant arithmetic R# admits finite models whose domains are the integers modulo n rather than the expected natural numbers. Less well appreciated is the fact that the logic of these models is much more subtle than that of the three-valued structure in which they are usually presented. In this paper we consider the DeMorgan monoids in which R# can be modelled, deriving a fairly complete account of those modelling the stronger arithmetic RM# modulo n and a partial account for the case of R# modulo a prime number. The more general case in which the modulus is arbitrary is shown to lead to infinite propositional structures even with the additional constraint that '0=1' implies everything

Epimorphisms in varieties of subidempotent residuated structures

A commutative residuated lattice A is said to be subidempotent if the lower bounds of its neutral element e are idempotent (in which case they naturally constitute a Brouwerian algebra A*). It is proved here that epimorphisms are surjective in a variety K of such algebras A (with or without involution), provided that each finitely subdirectly irreducible algebra B in K has two properties: (1) B is generated by lower bounds of e, and (2) the poset of prime filters of B* has finite depth. Neither (1) nor (2) may be dropped. The proof adapts to the presence of bounds. The result generalizes some recent findings of G. Bezhanishvili and the first two authors concerning epimorphisms in varieties of Brouwerian algebras, Heyting algebras and Sugihara monoids, but its scope also encompasses a range of interesting varieties of De Morgan monoids

Varieties of De Morgan monoids : covers of atoms

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 nitely subdirectly irreducible De Morgan monoids may be mapped by non-injective homomorphisms. The homomorphic pre-images 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 V(C4) within U are revealed here. There are just ten of them (all nitely 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 nite subdirectly irreducible algebra is projective. Beyond U, all covers of V(C4) [or of V(D4)] within DMM are discriminator varieties. Of these, we identify in nitely many that are nitely generated, and some that are not. We also prove that there are just 68 minimal quasivarieties of De Morgan monoids.The European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant, RVO 67985807 and by the CAS-ICS postdoctoral fellowship, the National Research Foundation of South Africa and DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa.https://www.cambridge.org/core/journals/review-of-symbolic-logic2021-06-01am2021Mathematics and Applied Mathematic

Singly generated quasivarieties and residuated structures

Please read abstract in the article.H2020 Marie Skłodowska-Curie Actions; DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa and National Research Foundation of South Africa.https://onlinelibrary.wiley.com/journal/15213870hj2021Mathematics and Applied Mathematic

Beberapa Operasi dan Sifat-Sifat Aljabar untuk N-Soft Set

Dalam artikel ini, difokuskan pada dua tujuan utama. Pertama, didefinisikan beberapa operasi biner dan non-biner pada N-soft set. Dalam operasi biner, dipelajari gabungan terbatas, gabungan diperluas, irisan terbatas dan irisan diperluas. Dalam operasi non-biner, tiga jenis komplemen dipelajari. Dibuktikan hukum De Morgan mengenai komplemen teratas dan komplemen bawah untuk N-soft set dimana N tetap dan diberikan contoh untuk menunjukkan bahwa hukum De Morgan tidak berlaku jika kita mengambil N yang berbeda. Yang kedua, dibuktikan beberapa sifat-sifat aljabar yang berlaku terkait dengan operasi-operasi yang didefinisikan. Dipelajari koleksi N-soft set yang berbeda menjadi monoid komutatif idempoten dan akibatnya menunjukkan, bahwa monoid menimbulkan hemiring N-soft set. Beberapa hemiring ini berubah menjadi lattice. Akhirnya, ditunjukkan koleksi semua N-soft set dengan himpunan parameter penuh E. Kata Kunci : N-soft set, Struktur Aljabar, Komplemen Teratas, Komplemen Bawah, Hukum De Morgan, Monoid, Komutatif, Idempoten, Hemiring, Lattice

Structural completeness in relevance logics

It is proved that the relevance logic R (without sentential constants) has no structurally complete consistent axiomatic extension, except for classical propositional logic. In fact, no other such extension is even passively structurally complete.http://link.springer.com/journal/112252017-06-30hb201

The algebraic significance of weak excluded middle laws

Please read abstract in the article.National Research Foundation of South Africa; Ministry of Science and Innovation of Spain; Agència de Gestió d'Ajuts Universitaris i de Recerca.https://onlinelibrary.wiley.com/journal/15213870hj2023Mathematics and Applied Mathematic