Sectionally Pseudocomplemented Residual Lattice

Abstract

At first, we recall the basic concept, By a residual lattice is meant an algebra L = (L,∨,∧,∗,o,0,1) such that (i) L = (L,∨,∧,0,1) is a bounded lattice, (ii) L = (L,∗,1) is a commutative monoid, (iii) it satisfies the so-called adjoin ness property: (x ∨ y) ∗ z = y if and only if y ≤ z ≤ x o y Let us note [7] that x ∨ y is the greatest element of the set (x ∨ y) ∗ z = y Moreover, if we consider x ∗ y = x ∧ y , then x o y is the relative pseudo-complement of x with respect to y, i. e., for ∗ = ∧ residuated lattices are just relatively pseudo-complemented lattices. The identities characterizing sectionally pseudocomplemented lattices are presented in [3] i.e. the class of these lattices is a variety in the signature {∨,∧,o,1}. We are going to apply a similar approach for the adjointness property: Key words: Residuated lattice; non Distributive; Residuated Abeliean; commutative monoid: DOI: http://dx.doi.org/10.3329/diujst.v6i2.9345 DIUJST 2011; 6(2): 53-54