Around a central element of a nearlattice

Abstract

A nearlattice S is a meet semilattice together with the property that any two elements possessing a common upper bound have a supremum. It is well known that if n ? S is a neutral and upper element then its isotope S_n = (S; ?) is again a nearlattice, where x ? y = (x ? y) ? (x ? n) ? (y ? n) for all x, y ? S . In this paper we have discussed the central elements in a nearlattice and also in a lattice. We included several characterizations of these elements. We showed that for a central element n ? S, P_n(S) ? (n]^d × [n), where P_n(S) is the set of principal n-ideals of S. Then we proved that for a central element n ? S, an element t ? S is central if and only if it is central in S_n. We also proved that for a lattice L, L_n is again a lattice if and only if n is central. Finally we showed that B is a Boolean algebra if and only if B_n is a Boolean algebra with same complement when n is central. Moreover, B ? B_n.

DOI: http://dx.doi.org/10.3329/ganit.v31i0.10312

GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 31 (2011) 95-104