Sur les algèbres de Lukasiewicz injectives

“…Since the notion of SH-algebra of order n is a generalization of those of Boolean algebra, symmetrical Boolean algebra and three-valued Lukasiewicz algebra we infer the following facts: I n a direct manner we can show that h is a homomorphism from A t o C. Furthermore as in [38] it can be shown tha,t, if C is an injective SH-algebra of order n, then C must be complete and centered. Thus our problem is reduced to know the injective objects in the category of symmetrical Boolean algebras.…”

Section: Injective Sh-algebras Of Order Nmentioning

confidence: 89%

Symmetrical Heyting Algebras With Operators

Iturrioz

1983

Mathematical Logic Qtrly

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SYMMETRICAL HEYTING ALGEBRAS WITH OPERATORS by LUISA ITURRIOZ in Villeurbanne (France) S,x v i S , x = 1 , with i x = 5 3 0 . 3* LUISA ITUIhRIOZIt follows from the above definition and from the fact that the class of all symmetrical Heyting algebras is equationally definable ([30], [34]) that the class of all symmetrical Heyting algebras of order n is also equationally definable. We will refer to a SHalgebra A of order n, for short.A SH-algebra A of order n is said to be non-degenerate if it is a non-degenerate algebra, i.e. if it contains a t least two different elements. If (A, 0, 1, A, v, a , 1, -, S,) is a SH-algebra of order 2, then S,x = z for all 5 E A and (A, 0 , 1, A, v, a, i , -) is a symmetrical Boolean algebra ([20]), noted SB-algebra for short. We are going t o see later that SH-algebras of order 3 are equivalent to involutive three-valued Heyting algebras studied in [14] and [15]. 3.2.

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“…This condition has been introduced by Moisil in his 1940papers Moisil (1940, with the name of modal principle of excluded middle, in order to give an algebraic semantic of Lukasiewicz three valued logic, further on investigated in the period 1963-65 by Monteiro in Monteiro (1963, 1965, (and see also the Cignoli and Monteiro contribution Cignoli and Monteiro (1965)) and successively by Becchio in the 1973 paper Becchio (1973) (for a treatment in the BZ poset context see the 1989 paper Cattaneo and Nisticò (1989), in particular condition (ne-1) of sect. 3).…”

Section: 2mentioning

confidence: 99%

Algebraic models of deviant modal operators based on de Morgan and Kleene lattices

Cattaneo

Ciucci

Dubois

2011

Information Sciences

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Abstract. An algebraic model of a kind of modal extension of de Morgan logic is described under the name MDS5 algebra. The main properties of this algebra can be summarized as follows: (1) it is based on a de Morgan lattice, rather than a Boolean algebra; (2) a modal necessity operator that satisfies the axioms N , K, T , and 5 (and as a consequence also B and 4) of modal logic is introduced; it allows one to introduce a modal possibility by the usual combination of necessity operation and de Morgan negation; (3) the necessity operator satisfies a distributivity principle over joins. The latter property cannot be meaningfully added to the standard Boolean algebraic models of S5 modal logic, since in this Boolean context both modalities collapse in the identity mapping. The consistency of this algebraic model is proved, showing that usual fuzzy set theory on a universe U can be equipped with a MDS5 structure that satisfies all the above points (1)- (3), without the trivialization of the modalities to the identity mapping. Further, the relationship between this new algebra and Heyting-Wajsberg algebras is investigated. Finally, the question of the role of these deviant modalities, as opposed to the usual nondistributive ones, in the scope of knowledge representation and approximation spaces is discussed.

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Sur les algèbres de Lukasiewicz injectives

Cited by 7 publications

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NUMBER OF EPIMORPHISMS BETWEEN FINITE AXLED n-VALUED MOISIL ALGEBRAS

NUMBER OF EPIMORPHISMS BETWEEN FINITE AXLED n-VALUED MOISIL ALGEBRAS

Symmetrical Heyting Algebras With Operators

Algebraic models of deviant modal operators based on de Morgan and Kleene lattices

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