889 research outputs found
Homology of Distributive Lattices
We outline the theory of sets with distributive operations: multishelves and
multispindles, with examples provided by semi-lattices, lattices and skew
lattices. For every such a structure we define multi-term distributive homology
and show some of its properties. The main result is a complete formula for the
homology of a finite distributive lattice. We also indicate the answer for
unital spindles and conjecture the general formula for semi-lattices and some
skew lattices. Then we propose a generalization of a lattice as a set with a
number of idempotent operations satisfying the absorption law.Comment: 30 pages, 3 tables, 3 figure
Around the Hossz\'u-Gluskin theorem for -ary groups
We survey results related to the important Hossz\'u-Gluskin Theorem on
-ary groups adding also several new results and comments. The aim of this
paper is to write all such results in uniform and compressive forms. Therefore
some proofs of new results are only sketched or omitted if their completing
seems to be not too difficult for readers. In particular, we show as the
Hossz\'u-Gluskin Theorem can be used for evaluation how many different -ary
groups (up to isomorphism) exist on some small sets. Moreover, we sketch as the
mentioned theorem can be also used for investigation of
-independent subsets of semiabelian -ary groups for some
special families of mappings
Hindman's finite sums theorem and its application to topologizations of algebras
The first part of the paper is a brief overview of Hindman's finite sums
theorem, its prehistory and a few of its further generalizations, and a modern
technique used in proving these and similar results, which is based on
idempotent ultrafilters in ultrafilter extensions of semigroups. The second,
main part of the paper is devoted to the topologizability problem of a wide
class of algebraic structures called polyrings; this class includes Abelian
groups, rings, modules, algebras over a ring, differential rings, and others.
We show that the Zariski topology on such an algebra is always non-discrete.
Actually, a much stronger fact holds: if is an infinite polyring, a
natural number, and a map of into is defined by a term in
variables, then is a closed nowhere dense subset of the space
with its Zariski topology. In particular, is a closed nowhere dense
subset of . The proof essentially uses a multidimensional version of
Hindman's finite sums theorem established by Bergelson and Hindman. The third
part of the paper lists several problems concerning topologization of various
algebraic structures, their Zariski topologies, and some related questions.
This paper is an extended version of the lecture at Journ\'ees sur les
Arithm\'etiques Faibles 36: \`a l'occasion du 70\`eme anniversaire de Yuri
Matiyasevich, delivered on 7th July, 2017, in Saint Petersburg.Comment: The main result of the paper, Theorem 2.4.1, was proved around 2010
but not published until 2017 though presented at several seminars and
conferences, e.g. Colloquium Logicum 2012 in Paderborn, and included in
author's course lectured at the Steklov Mathematical Institute in 201
Free three-valued Closure Lukasiewicz Algebras
In this paper, the structure of finitely generated free objects in the variety of three-valued closure Lukasiewicz algebras is determined. We describe their indecomposable factors and we give their cardinality.Fil: Abad, Manuel. Universidad Nacional del Sur. Departamento de Matemática; ArgentinaFil: DÃaz Varela, José Patricio. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Centro CientÃfico Tecnológico Conicet - BahÃa Blanca. Instituto de Matemática BahÃa Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática BahÃa Blanca; ArgentinaFil: Rueda, Laura Alicia. Universidad Nacional del Sur. Departamento de Matemática; ArgentinaFil: SuardÃaz, Ana MarÃa. Universidad Nacional del Sur. Departamento de Matemática; Argentin
State morphism MV-algebras
We present a complete characterization of subdirectly irreducible MV-algebras
with internal states (SMV-algebras). This allows us to classify subdirectly
irreducible state morphism MV-algebras (SMMV-algebras) and describe single
generators of the variety of SMMV-algebras, and show that we have a continuum
of varieties of SMMV-algebras
Whitney algebras and Grassmann's regressive products
Geometric products on tensor powers of an exterior
algebra and on Whitney algebras \cite{crasch} provide a rigorous version of
Grassmann's {\it regressive products} of 1844 \cite{gra1}. We study geometric
products and their relations with other classical operators on exterior
algebras, such as the Hodge operators and the {\it join} and {\it meet}
products in Cayley-Grassmann algebras \cite{BBR, Stew}. We establish encodings
of tensor powers and of Whitney algebras in
terms of letterplace algebras and of their geometric products in terms of
divided powers of polarization operators. We use these encodings to provide
simple proofs of the Crapo and Schmitt exchange relations in Whitney algebras
and of two typical classes of identities in Cayley-Grassmann algebras
- …