Michał M. Stronkowski

EMBEDDING ENTROPIC ALGEBRAS INTO SEMIMODULES AND MODULES

An algebra is entropic if its basic operations are homomorphisms. The paper is focused on representations of such algebras. We prove the following theorem: An entropic algebra without constant basic operations which satisfies so called Szendrei identities and such that all its basic operations of arity at least two are surjective is a subreduct of a semimodule over a commutative semiring. Our theorem is a straightforward generalization of Ježeks and Kepkas theorem for groupoids. As a consequence we obtain that a mode (entropic and idempotent algebra) is a subreduct of a semimodule over a commutative semiring if and only if it satisfies Szendrei identities. This provides a complete solution to the problem in mode theory asking for a characterization of modes which are subreducts of semimodules over commutative semirings. In the second part of the paper we use our theorem to show that each entropic cancellative algebra is a subreduct of a module over a commutative ring. It extends a theorem of Romanowska and Smith about modes.

Rab geranylgeranyl transferase β subunit is essential for male fertility and tip growth in Arabidopsis

Summary Rab proteins are post-translationally geranylgeranylated by Rab geranylgeranyl transferase (RGT) αβ. Mutations in each of the RGTB genes cause a tip growth defect whereas the double mutant is male sterile.

Almost structural completeness; an algebraic approach

Abstract A deductive system is structurally complete if all of its admissible inference rules are derivable. For several important systems, like the modal logic S5, failure of structural completeness is caused only by the underivability of a passive rule, i.e., a rule whose premise is not unifiable by any substitution. Neglecting passive rules leads to the notion of almost structural completeness, that means, to the derivability of admissible non-passive rules. We investigate almost structural completeness for quasivarieties and varieties of general algebras. The results apply to all algebraizable deductive systems. Firstly, various characterizations of almost structurally complete quasivarieties are presented. Two of them are general: the one expressed with finitely presented algebras, and the one expressed with subdirectly irreducible algebras. The next one is restricted to quasivarieties with the finite model property and equationally definable principal relative congruences, where the condition is verifiable on finite subdirectly irreducible algebras. Some connections with exact and projective unification are included. Secondly, examples of almost structurally complete varieties are provided. Particular emphasis is put on varieties of closure algebras, that are known to constitute adequate semantics for normal extensions of the modal logic S4. A certain infinite family of such almost structurally complete, but not structurally complete, varieties is constructed. Every variety from this family has a finitely presented unifiable algebra which does not embed into any free algebra for this variety. Hence unification is not unitary there. This shows that almost structural completeness is strictly weaker than projective unification for varieties of closure algebras.

On structural completeness versus almost structural completeness problem: A discriminator varieties case study

Fil: Campercholi, Miguel Alejandro Carlos. Universidad Nacional de Cordoba. Facultad de Matematica, Astronomia y Fisica; Argentina

Quasivarieties with Definable Relative Principal Subcongruences

For quasivarieties of algebras, we consider the property of having definable relative principal subcongruences, a generalization of the concepts of definable relative principal congruences and definable principal subcongruences. We prove that a quasivariety of algebras with definable relative principal subcongruences has a finite quasiequational basis if and only if the class of its relative (finitely) subdirectly irreducible algebras is strictly elementary. Since a finitely generated relatively congruence-distributive quasivariety has definable relative principal subcongruences, we get a new proof of the result due to D. Pigozzi: a finitely generated relatively congruence-distributive quasivariety has a finite quasi-equational basis.

Remarks on Smooth Real-Compactness for Sikorski Spaces

Abstract It is known that every Sikorski space with the countably generated differential structure is smoothly real-compact. It means that every homomorphism from its differential structure, which forms a ring of smooth real-valued functions into the ring of real numbers, is an evaluation. This result is sharp: there is a non-smoothly real-compact Sikorski space with the differential structure which is not countably generated. We provide an easy example demonstrating this. By modifying this example we are able to show a certain shortcoming of the generator embedding, comparing to the canonical embedding, for Sikorski spaces. Finally, we note that a homomorphism from the ring of smooth functions of a Sikorski space into the ring of real numbers is an evaluation if and only if it is continuous.

Embedding sums of cancellative modes into functorial sums

Abstract The paper discusses a representation of modes (idempotent and entropic algebras) as subalgebras of so-called functorial sums of cancellative algebras. We show that each mode that has a homomorphism onto an algebra satisfying a certain additional condition, with corresponding cancellative congruence classes, embeds into a functorial sum of cancellative algebras. We also discuss typical applications of this result.