arXiv subjects

Alejandro Petrovich

Publications and source records attributed to Alejandro Petrovich.

4 recordsLinked to original sources

arXiv2024

Definability of band structures on posets

The idempotent semigroups (bands) that give rise to partial orders by defining $a \leq b \iff a \cdot b = a$ are the "right-regular" bands (RRB), which are axiomatized by $x\cdot y \cdot x = y \cdot x$. In this work we consider the class of "associative posets", which comprises all partial orders underlying right-regular bands, and study to what extent the ordering determines the possible "compatible" band structures and their canonicity. We show that the class of associative posets in the signature $\{ \leq \}$ is not first-order axiomatizable. We also show that the Axiom of Choice is equivalent over $\mathit{ZF}$ to the fact that every tree with finite branches is associative. We study the smaller class of "normal" posets (corresponding to right-normal bands) and give a structural characterization.

math.LO↗

arXiv2017

Fans in the Theory of Real Semigroups II. Combinatorial Theory

In the paper: Fans in the Theory of Real Semigroups. I. Algebraic Theory (submitted) we introduced the notion of fan in the categories of real semigoups and their dual abstract real spectra and developed the algebraic theory of these structures. In this paper we develop the combinatorial theory of ARS-fans, i.e., fans in the dual category of abstract real spectra. Every ARS is a spectral space and hence carries a natural partial order called the {\it specialization partial order}. Our main result shows that the isomorphism type of a finite fan in the category ARS is entirely determined by its order of specialization. The main tools used to prove this result are: (1) Crucial use of the theory of {\it ternary semigroups}, a class of semigroups underlying that of RSs; (2) Every ARS-fan is a disjoint union of abstract order spaces (called {\it levels}); (3) Every level carries a natural involution of abstract order spaces, and (4) The notion of a {\it standard generating system}, a combinatorial tool replacing, in the context of ARSs, the (absent) tools of combinatorial geometry (matroid theory) employed in the cases of fields and of abstract order spaces.

math.AG↗

arXiv2017

Fans in the Theory of Real Semigroups I. Algebraic Theory

In a previous paper we introduced the notion of a {\it real semigroup} (RS) as an axiomatic framework to study diagonal quadratic forms with arbitrary entries over (commutative, unitary) semi-real rings. Two important classes of RSs were studied at length in previous papers. In this paper we introduce and develop the algebraic theory of {\it RS-fans}, a third class of RSs providing a vast generalization of homonymous notions previously existing in field theory and in the theories of abstract order spaces and of reduced special groups; for a background on fans, see paragraph A of the Introduction, below. The contents of this paper are briefly reviewed in paragraph B of the Introduction. The combinatorial theory of the structures dual to RS-fans, called {\it ARS-fans}, is the subject of the paper: M. Dickmann, A. Petrovich, {\em Fans in the Theory of Real Semigroups. II. Combinatorial Theory}, 20 pp., submitted., a continuation of the present paper.

math.AG↗

arXiv2015

A duality for (n+1)-valued MV-algebras

MV-algebras were introduced by Chang to prove the completeness of the infinite-valued Lukasiewicz propositional calculus. In this paper we give a categorical equivalence between the varieties of (n+1)-valued MV-algebras and the classes of Boolean algebras endowed with a certain family of filters. Another similar categorical equivalence is given by A. Di Nola and A. Lettieri. Also, we point out the relations between this categorical equivalence and the duality established by R. Cignoli, which can be derived from results obtained by P. Niederkorn on natural dualities for varieties of MV-algebras.

math.LO↗