4,143 research outputs found
Admissibility via Natural Dualities
It is shown that admissible clauses and quasi-identities of quasivarieties
generated by a single finite algebra, or equivalently, the quasiequational and
universal theories of their free algebras on countably infinitely many
generators, may be characterized using natural dualities. In particular,
axiomatizations are obtained for the admissible clauses and quasi-identities of
bounded distributive lattices, Stone algebras, Kleene algebras and lattices,
and De Morgan algebras and lattices.Comment: 22 pages; 3 figure
Checking Admissibility Using Natural Dualities
This paper presents a new method for obtaining small algebras to check the
admissibility-equivalently, validity in free algebras-of quasi-identities in a
finitely generated quasivariety. Unlike a previous algebraic approach of
Metcalfe and Rothlisberger that is feasible only when the relevant free algebra
is not too large, this method exploits natural dualities for quasivarieties to
work with structures of smaller cardinality and surjective rather than
injective morphisms. A number of case studies are described here that could not
be be solved using the algebraic approach, including (quasi)varieties of
MS-algebras, double Stone algebras, and involutive Stone algebras
The variety of quasi-Stone algebras does not have the amalgamation property
We give an example showing that the variety of quasi-Stone algebras does not have the amalgamation propert
Reticulation of Quasi-commutative Algebras
The commutator operation in a congruence-modular variety allows
us to define the prime congruences of any algebra and the
prime spectrum of . The first systematic study of this spectrum
can be found in a paper by Agliano, published in Universal Algebra (1993).
The reticulation of an algebra is a bounded distributive
algebra , whose prime spectrum (endowed with the Stone topology) is
homeomorphic to (endowed with the topology defined by Agliano). In a
recent paper, C. Mure\c{s}an and the author defined the reticulation for the
algebras in a semidegenerate congruence-modular variety ,
satisfying the hypothesis : the set of compact congruences of
is closed under commutators. This theory does not cover the Belluce
reticulation for non-commutative rings. In this paper we shall introduce the
quasi-commutative algebras in a semidegenerate congruence-modular variety
as a generalization of the Belluce quasi-commutative rings. We
define and study a notion of reticulation for the quasi-commutative algebras
such that the Belluce reticulation for the quasi-commutative rings can be
obtained as a particular case. We prove a characterization theorem for the
quasi-commutative algebras and some transfer properties by means of the
reticulationComment: arXiv admin note: text overlap with arXiv:2205.0217
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