Sectionally Semicomplemented Distributive Nearlattices

Abstract

A nearlatticeS is a meet semilattice together with the property that any two elements possessing a common upper bound have a supremum. Here the authors have introduced the notion of sectionally semicomplemented distributive nearlattices and given several characterizations of them. The skeleton SCon(S) of Con(S), the congruence lattice, consists of all those nearlattice congruences which are the pseudocomplements of members of Con(S). The relationship between skeletal congruences and kernel of skeletal congruences leads to numerous characterizations of sectionally semicomplemented distributive nearlattices and semiboolean algebras. For example we prove, for a distributive nearlattice S with 0, the following conditions are equivalent: (i) S is sectionally semicomplemented (ii) The map Θ Θ ker Θ of SCon(S) onto KSCon(S) is one-to-one. (iii) The map Θ Θ ker Θ of SCon(S) onto KSCon(S) preserves finite joins. (iv) The map Θ Θ ker Θ is a lattice isomorphism of SCon(S) onto KSCon(S), whose inverse is the map J Θ(J)**.