T-algebras of the monad L-fuzz. (English) Zbl 0591.18005

Machner, Joachim

Czech. Math. J. 35(110), 515-528 (1985).

Author’s summary: ”The main purpose of this article is the characterization of T-algebras (A,h) of the monad L-Fuzz, which is connected with the fuzzification of mathematical objects, expecially automata, applying a brouwerian lattice L. The power set monad is a submonad of L-Fuzz and, as is well known, its T-algebras are precisely the complete sup-semi-lattices. In the case of T-algebras of L-Fuzz the set A will also have the complete lattice structure making it possible to construct a Galois correspondence (h,g) between TA and the dual \(A^ d\) of A. The above mentioned characterization will be performed by the statement of four independent conditions on g to be the residuated map of the morphism h of a T-algebra for L-Fuzz.
In Section 1, basic facts on the Kleisli and Eilenberg-Moore constructions are summarized and the monad L-Fuzz is constituted. Section 2 starts with a partial order on the underlying set A of the T-algebra (A,h), which is shown to be a complete lattice order. Having introduced the Galois correspondence (h,g) additional properties of g are established, a suitable selection of which will be characteristic. Section 3 studies the independence of the characteristic conditions obtained in the preceding section, while the last section is supplementary and contains some applications.”

Reviewer: M.Eytan

MSC:

18C20	Eilenberg-Moore and Kleisli constructions for monads
03E72	Theory of fuzzy sets, etc.
06A15	Galois correspondences, closure operators (in relation to ordered sets)
06D20	Heyting algebras (lattice-theoretic aspects)
06B23	Complete lattices, completions
18C15	Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads
18B20	Categories of machines, automata

Keywords:

Kleisli category; fuzzy sets; Galois connection; monad; brouwerian lattice; Galois correspondence; T-algebra; complete lattice

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