153 research outputs found
Stone-type representations and dualities for varieties of bisemilattices
In this article we will focus our attention on the variety of distributive
bisemilattices and some linguistic expansions thereof: bounded, De Morgan, and
involutive bisemilattices. After extending Balbes' representation theorem to
bounded, De Morgan, and involutive bisemilattices, we make use of Hartonas-Dunn
duality and introduce the categories of 2spaces and 2spaces. The
categories of 2spaces and 2spaces will play with respect to the
categories of distributive bisemilattices and De Morgan bisemilattices,
respectively, a role analogous to the category of Stone spaces with respect to
the category of Boolean algebras. Actually, the aim of this work is to show
that these categories are, in fact, dually equivalent
On Implicator Groupoids
In a paper published in 2012, the second author extended the well-known fact
that Boolean algebras can be defined using only implication and a constant, to
De Morgan algebras-this result led him to introduce, and investigate (in the
same paper), the variety I of algebras, there called implication zroupoids
(I-zroupoids) and here called implicator gruopids (I- groupoids), that
generalize De Morgan algebras. The present paper is a continuation of the paper
mentioned above and is devoted to investigating the structure of the lattice of
subvarieties of I, and also to making further contributions to the theory of
implicator groupoids. Several new subvarieties of I are introduced and their
relationship with each other, and with the subvarieties of I which were already
investigated in the paper mentioned above, are explored.Comment: This paper, except the appendix, will appear in Algebra Universalis.
25 pages, 4 figures, a revised version with a new titl
Some geometrical methods for constructing contradiction measures on Atanassov's intuitionistic fuzzy sets
Trillas et al. (1999, Soft computing, 3 (4), 197–199) and Trillas and Cubillo (1999, On non-contradictory input/output couples in Zadeh's CRI proceeding, 28–32) introduced the study of contradiction in the framework of fuzzy logic because of the significance of avoiding contradictory outputs in inference processes. Later, the study of contradiction in the framework of Atanassov's intuitionistic fuzzy sets (A-IFSs) was initiated by Cubillo and Castiñeira (2004, Contradiction in intuitionistic fuzzy sets proceeding, 2180–2186). The axiomatic definition of contradiction measure was stated in Castiñeira and Cubillo (2009, International journal of intelligent systems, 24, 863–888). Likewise, the concept of continuity of these measures was formalized through several axioms. To be precise, they defined continuity when the sets ‘are increasing’, denominated continuity from below, and continuity when the sets ‘are decreasing’, or continuity from above. The aim of this paper is to provide some geometrical construction methods for obtaining contradiction measures in the framework of A-IFSs and to study what continuity properties these measures satisfy. Furthermore, we show the geometrical interpretations motivating the measures
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