Algebra Universalis最新文献

A finite semigroup is finitely related (has finite degree) if its term functions are determined by a finite set of finitary relations. For example, it is known that all nilpotent semigroups are finitely related. A nilpotent monoid is a nilpotent semigroup with adjoined identity. We show that every 4-nilpotent monoid is finitely related. We also give an example of a 5-nilpotent monoid that is not finitely related. To our knowledge, this is the first example of a finitely related semigroup where adjoining an identity yields a semigroup which is not finitely related. We also provide examples of finitely related semigroups which have subsemigroups, homomorphic images, and in particular Rees quotients, that are not finitely related.

C. Greene introduced the shuffle lattice as an idealized model for DNA mutation and discovered remarkable combinatorial and enumerative properties of this structure. We attempt an explanation of these properties from a lattice-theoretic point of view. To that end, we introduce and study an order extension of the shuffle lattice, the bubble lattice. We characterize the bubble lattice both locally (via certain transformations of shuffle words) and globally (using a notion of inversion set). We then prove that the bubble lattice is extremal and constructable by interval doublings. Lastly, we prove that our bubble lattice is a generalization of the Hochschild lattice studied earlier by Chapoton, Combe and the second author.

We show that for a large class of varieties of algebras, the equational theory of the congruence lattices of the members is not finitely based.

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 aim of this article is to develop a Priestley-style duality for the variety of DN-algebras. In order to achieve this, we use the concept of free distributive lattice extension of a DN-algebra. We establish a connection with the Priestley duality for distributive lattices. Finally, we present topological descriptions for the lattice of filters, for the lattice of congruences, and for certain kinds of subalgebras of a DN-algebra.

The present paper deals with (L_0)-valued measures and the associated Orlicz–Kantorovich lattice. The considered Orlicz–Kantorovich lattice, endowed with the Luxemburg norm, is represented as a measurable bundle of classical Orlicz spaces associated with scalar measures. This kind of representation allows us to investigate positive contractions and apply the corresponding zero-two laws on the classical Orlicz spaces, to prove vector versions of “zero-two” laws on the considered Orlicz–Kantorovich space.

Orrin Frink in 1962 stated incorrectly, that there exist free pseudocomplemented meet-semilattices, which are not lattices. This is disproved in the present paper. We have shown that any free PCS is in fact a sectionally pseudocomplemented lattice.

The 93 minions of Boolean functions stable under left composition with the clone of self-dual monotone functions are described. As an easy consequence, all ((C_1,C_2))-clonoids of Boolean functions are determined for an arbitrary clone (C_1) and for any clone (C_2) containing the clone of self-dual monotone functions.

We investigate in this article regular Heyting algebras by means of Esakia duality. In particular, we give a characterisation of Esakia spaces dual to regular Heyting algebras and we show that there are continuum-many varieties of Heyting algebras generated by regular Heyting algebras. We also study several logical applications of these classes of objects and we use them to provide novel topological completeness theorems for inquisitive logic, (texttt{DNA})-logics and dependence logic.