PROPOSITIONAL LOGIC, FRAMES, AND FUZZY ALGEBRA
作者:
B. Banaschewski,
期刊:
Quaestiones Mathematicae
(Taylor Available online 1999)
卷期:
Volume 22,
issue 4
页码: 481-508
ISSN:1607-3606
年代: 1999
DOI:10.1080/16073606.1999.9632100
出版商: Taylor & Francis Group
关键词: 03G10;03E72
数据来源: Taylor
摘要:
This paper offers a new look at such things as the fuzzy subalgebras and congruences of an algebra, the fuzzy ideals of a ring or a lattice, and similar entities, by exhibiting them as the models, in the chosen frameTof truth values, of naturally corresponding propositional theories. This provides a systematic approach to the study of the partially ordered sets formed by these various entities, and we demonstrate its usefulness by employing it to derive a number of results, some old and some new, concerning these partially ordered sets. In particular, we prove they are complete lattices, algebraic or continuous, depending on whether T is algebraic or continuous, respectively (Proposition 3); they satisfy the same lattice identities for arbitraryTthat hold in the caseT= 2 (Corollary of Proposition 4); and they are coherent frames for any coherentTwhenever this is the case forT= 2 (Proposition 6). In addition we show, generalizing a result by Makamba and Murali [10], that the familiar classical situations where the congruences of an algebra correspond to certain other entities, such as the normal subgroups of a group or the ideals of a ring, extend to the fuzzy case by proving that the corresponding propositional theories are equivalent (Proposition 2). Further, we obtain the result of Gupta and Kantroo [5] that the fuzzy radical ideals of a commutative ring with unit are the meets of fuzzy prime ideals for arbitrary continuous T in place of the unit interval, using basic facts concerning continuous frames (Proposition 7).
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