Remark on the Unital Quantale Q[e]

Abstract

In this paper, we investigate some properties of the unital quantale Q[e], in terms of the quantic quotients of Q[e] we study the extensions of quantic nuclei of Q to Q[e] and prove that for any non-trivial quantale Q, Q[e] is not simple. Also, we give some applications for the unital quantale Q[e].Keywords Quantale · Quantic quotient · Quantic nucleus · Simple quantale Mathematics Subject Classification (2010) 06F07 PreliminariesQuantales were introduced in [8] by C.J. Mulvey in order to provide a lattice theoretic setting for studying non-commutative C * -algebras, as well as a constructive foundation for quantum mechanics. The study of such partially ordered algebraic structures goes back to a series of papers by M. Ward and P. R. Dilworth [2,17,18] in the 1930s. Following C.J. Mulvey, various types and aspects of quantales have been considered by many researchers [5][6][7][9][10][11][12][13][14][15][16]. Simple quantale, spatial quantale and idempotent quantale are very important classes of quantales and they have been studied systemically in [5,7,10,12,15,16]

New constructions of uni-nullnorms on certain classes of bounded lattices by closure (interior) operators

“…The theory of representations of Boolean algebras [49] has shown that three mother structures when combining together can create some interesting and meaningful results. Partial researches can be seen in [2,28,30,45,50,[53][54][55][56]64,65]. In this article I mainly use uninorms combining a t-norm T (a t-conorm S) and closure operators (interior operators) to construct uni-nullnorms on certain special classes of bounded lattices.…”

“…[20,30,45]) Let L be a bounded lattice and a, b ∈ L be given, where a ≤ b. Then the mapping cl : L → L defined by the following are all closure operators: for arbitrary p ∈ L, (i) cl(p) = p; (ii) cl(p) = 1; (iii) cl(p) = p∨a; (iv) cl(p) = a → p;(v) cl(p) = (p → a) → a; (vi) (f, g) is a Galois connection, cl = g • f .…”