Algebra Universalis最新文献

Extending Sparks’s theorem, we determine the cardinality of the lattice of ((C_1,C_2))-clonoid of Boolean functions in the cases where the target clone (C_2) is the clone of projections. Moreover, we explicitly describe the ((C_1,C_2))-clonoid of Boolean functions in the cases where the source clone (C_1) is one of the four clones of monotone functions or contains the discriminator function.

Płonka sums are a powerful technique for the representation of algebras in regular varieties. However, certain representations of algebras in irregular varieties—like Polin’s variety or the variety of pseudocomplemented semilattices—bear striking similarities to Płonka sums, although they differ from them in some important respects. We aim at finding a convenient umbrella under which these constructions, as well as other ones of a similar kind, can be subsumed. Inspired by Grätzer and Sichler’s work on Agassiz sums, we appropriately enrich the structure of semilattice direct systems and we modify the attendant definition of a sum, while still encompassing Płonka sums as a special case. We prove that the above-mentioned representations of Polin algebras and pseudocomplemented semilattices can be recast in terms of this new framework. Finally, we investigate the problem as to which identities are preserved by the construction.

A retract variety is defined as a class of algebras closed under isomorphisms, retracts and products. Let a principal retract variety be generated by one algebra and a set-principal retract variety be generated by some set of algebras. It is shown that (a) not each set-principal retract variety is principal, and (b) not each retract variety is set-principal. A class of connected monounary algebras ({mathcal {S}}) such that every retract variety of monounary algebras is generated by algebras that have all connected components from ({mathcal {S}}) and at most two connected components are isomorphic is defined, this generating class is constructively described. All set-principal retract varieties of monounary algebras are characterized via degree function of monounary algebras.

In this paper, we show that every monadic ortholattice is isomorphic to a functional one, thereby resolving a recent question posed by Harding. We then study certain substitution-free reducts of the polyadic ortholattices, which we call locally finite (sigma )-free polyadic ortholattices, and demonstrate that they stand in a one-to-one correspondence with the locally finite diagonal-free cylindric ortholattices.

We study the finite basis problem for 4-element additively idempotent semirings whose additive reducts have the least element and two coatoms. Up to isomorphism, there are 93 such algebras. We show that with the exception of the semiring (S_{(4, 435)}), all of them are finitely based.

In this paper, we present a Priestley-type topological representation for (prec )-distributive (vee )-predomains, thereby answering an open problem posed by T. Bice. Moreover, we establish a dual equivalence between the category of (prec )-distributive (vee )-predomains with (prec )-morphisms and that of DP-compact pospaces with DP-morphisms. In particular, our results restrict to Hansoul-Poussart duality for bounded distributive sup-semilattices and to a Priestley duality for continuous frames.

A subset X of a groupoid is said to be deficient if (|X cdot X|le |X|). It is well-known that the probability that a random groupoid has a deficient t-element set with (tge 3) is zero. However, it is believed that the probability is not zero for 2-element sets. We prove that it is indeed not null and calculate the exact value. We explore some generalisations on deficient sets and their likelihoods.

As a main result, we characterize prime spectra of abelian lattice ordered groups. Further we introduce some categories based on spectral spaces, lattices and Priestley spaces. Then we have a characterization of the variety generated by the Chang MV-algebra and we study this variety. Next we generalize the results to every variety generated by a Komori chain.

Semirings of partial Boolean-valued functions arise in Bishop’s approach to constructive measure theory. In this paper, we treat such semirings axiomatically. Lifting the corresponding result for Boolean rings, we prove a representation theorem à la Stone that aligns with the intended semantics. Moreover, among the semirings at hand, we determine the order of those that are free over finitely many generators.