On a Problem of Garrett Birkhoff and related topics [MatSciRep: 94]

Abstract

This report forms part of the lectures given by the author on his own research work during his stay at MATSCIENCE as a Senior Research Fellow. Chapter l deals with the characterisation of the direct product of a Boolean ring and a lattice ordered group, thereby solving Birkhoff's problem no.l05 "Lattice Theory" A.M.S. Col. Pub. XXV (1948) (Is there a common abstraction which includes of Boolean rings (algebras) and lattice ordered groups as special cases?). These results which are also a generalisation of the author's paper "On a common abstraction of Boolean rings and lattice ordered groups I", Manatshefte fur Mathematik 73 411-421 (1969) have been accepted for publication in Math. Slovaca. In Chapter 2 "Dually residuated lattice ordered semi groups" or briefy D.R.L. semi group have been generalised to semi-dually residuated lattice ordered semigroups to include semi-Brouwerian algebras. Semi D.R.L. semigroups have many interesting properties of D.R.L. semigroups and these have been studied in detail. The relationship between a semi D.R.L. semi group and a Boolean-l-algebra has been discussed with some interesting results. The results of this Chapter have appeared as a note in Math. Seminar notes Vol.6 (1978). Chapter 3 is devoted to the study of the structure of a D.R.L. semigroup and a class of D.R.L. semi groups which can be obtained as a global sections with compact Casmiss of a sheaf of nontrivial totally ordered D.R.L. semigroups over a Boolean space, has been characterised by means of two conditions. These results have been communicated to Math. Seminar Notes. In the last chapter an attempt has been made to obtain the relationship between normal and distributive * lattices.