5 research outputs found
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
Semilinear De Morgan monoids and epimorphisms
DATA AVAILABILITY : Data sharing not applicable to this article as datasets were neither generated nor analysed.A representation theorem is proved for De Morgan monoids that are (i) semilinear, i.e., subdirect products of totally ordered algebras, and (ii) negatively generated, i.e., generated by lower bounds of the neutral element. Using this theorem, we prove that the De Morgan monoids satisfying (i) and (ii) form a variety—in fact, a locally finite variety. We then prove that epimorphisms are surjective in every variety of negatively generated semilinear De Morgan monoids. In the process, epimorphism-surjectivity is established for several other classes as well, including the variety of all semilinear idempotent commutative residuated lattices and all varieties of negatively generated semilinear Dunn monoids. The results settle natural questions about Beth-style definability for a range of substructural logics.The Operational Programme Research,
Development and Education of the Ministry of Education, Youth and Sports of the Czech
Republic, the EU and in part by the
National Research Foundation of South Africa. Open access funding provided by University of Pretoria.https://link.springer.com/journal/12hj2024Mathematics and Applied MathematicsNon
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
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