We present a uniform method of density elimination for several semilinear substructural logics. Especially, the density elimination for the involutive uninorm logic IUL is proved. Then the standard completeness of IUL follows as a lemma by virtue of previous work by Metcalfe and Montagna.
In this paper, we solve a long-standing open problem in the field of fuzzy logics, that is, the standard completeness for the involutive uninorm logic IUL. In fact, we present a uniform method of density elimination for several semilinear substructural logics. Especially, the density elimination for IUL is proved. Then the standard completeness for IUL follows as a lemma by virtue of previous work by Metcalfe and Montagna.
Our method for density elimination is generalized to the non-commutative substructurallogic GpsUL*. Then, the standard completeness of HpsUL* follows as a lemma by virtue of previouswork by Metcalfe and Montagna. This result shows that HpsUL* is the logic of pseudo-uninormsand their residua and answered the question posed by Prof. Metcalfe, Olivetti, Gabbay and Tsinakis.
For any odd positive integer x, define (x n ) n⩾0 and (a n ) n⩾1 of all 2 an is called the E-sequence of x. For example, the orbit and the E-sequence of 3 are (3, 5, 1, 1, . . .) and (1, 4, 2, 2, . . .), respectively.Given any sequence (a n ) n⩾1 of positive integers, if it is the E-sequence of the odd positive integer x, it is called to be convergent to x and, denoted by Ω − lim a n = x; if (a n ) n⩾1 is not the E-sequence of any odd positive integer, 2010 Mathematics Subject Classification. Primary 11A99; Secondary 11B83.
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