Around a central element of a nearlattice
Keywords:Central element, Nearlattice, Isotope, Boolean algebra
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 Sn = (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, Pn(S) ? (n]d × [n), where Pn(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 Sn. We also proved that for a lattice L, Ln is again a lattice if and only if n is central. Finally we showed that B is a Boolean algebra if and only if Bn is a Boolean algebra with same complement when n is central. Moreover, B ? Bn.
GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 31 (2011) 95-104
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