A Study of Sectionally Pseudocomplemented Lattice & Boolean Algebra

Abstract

This thesis studies the nature of a sectionally pesudocomplemented lattice and Boolean Algebra. Lattice theory is a part of Mathematics. Boolean algebra and Boolean function is an important of lattice theory. A nonempty set P together with a binary relation R is said to form a partially order set or a poset if the following conditiois hold: (i) Reflexivity (ii) Anti-symmetry (iii) Transitivity. A poset (L, ≤) is said to form a lattice if for every a, b ε L if a˅b and a ˄ b exist in L. A lattice is said to be complemented lattice if every element has a complement. Let L be a bounded distributive lattice, let a ε L an element a* ε L is called a pseudocomplement of a in L if the following conditions holds: (i) a ˄ a* = 0 (ii) ∀ x ε L, a ˄ x =0 implies that x ≤ a* . A complement distributive lattice is called a Boolean lattice. Since complements are unique in a Boolean lattice as an algebra with two binary operations A and v and one unary operation '.Boolean lattices so considered are called Boolean algebra. Moreover we can discuss on relatively pseudocomplemented Lattices. In this thesis, we have given several results on seetionnally (relatively) pseudocomplemented lattices which certainly extended and generalized many results in lattice theory. In chapter one is to outline and fix the notation for some of the concepts of lattices which are basic to this thesis. Some more definitions and formulate results on a orbitrary lattices for later use. We have considered this section as the base and background for the study of subsequent sections. For the background material in Lattice theory we have refered the readers to the of G. Birkoff [14] G. Gratzer [15] and V.K. Khanna [24] and several authors. In chapter two, we have given a description of difference classes of lattices. We have also studied normal lattices and distributive quasi-complemented lattices. Generalized stone lattices have been studied by H. Lakser [16,17], K.B Lee [20] and many other authors. We have given a characterization of minimal prime ideals of a sectionally pseudocomplemented lattices. Then we have shown that a distributive Lattice L with 0 is generalized stone if and only if it is both normal and sectionally quasi-complemented. In chapter three introduces the concept of relative annihilators in lattices. Relative annihilators in lattices were studied by several authors including Mandelker [21] and Verlet [22]. B.A. Davey [1] has used the annihiiators in studying relatively normal lattices. Here we have studied the relative annihilators in lattices. In terms of relative annihilators, we have characterized modular and distributive lattices. Relatively stone lattices have been studied by several authors including Mandelker [26], T.P. Speed [23] Gratzer and Schmidt [15]. Here we use given several characterizations of relatively stone lattices, which are certainly the generalization of above authors work. We have also shown for a distributive lattice L in which every closed interval in pseudocomplemented is relatively stone if and only if any two incomparable prime ideals of L are comaximal. In chapter four, we have studied lattices with the greatest element 1 where on each interval [a,1] an antitone bijection is defined. We have characterized these lattices by means of two induced binary operations proving that the resulting algebras form a variety. The congruence properties of this variety and the properties of the underlying lattices are investigated. We have shown that this variety contains a single minimal subquasi variety join-lattices, whose principal filters are Boolean lattices, were used by J.C. Abbott [13]. We have introduced a further generalization of this concept, defining the notion of a lattice with sectionally antitone bijection. We have also introduced Residuated Lattices studied by Ward and Diworth [26] and several authors. Two mono graph contain a compendium on residuated lattices. They are that by Blyth and Janowitz [2]. In this paper we will compare a certain modification of a residuated Lattice. In chapter five, It is shown that every directoid equipped with sectionally switching mappings can be represented as a certain implication algebra. The concept of directoid was introduced by J. Jezek and R. Quackenbush [19] in the sake to axiomatize algebraic structures defined by on upward directed ordered set. In certain sense, directoids generalize semilattices. In chapter six. We have studied switching Mapping introduced by Chajda and Emanovsky [3]. A mapping f of [a, 1] onto itself is called switching mapping if f(a) =1 and for x ε [a,1], a ≠x ≠1 in the section[q,1] is determined by that of [p,l], we say that the compatibility condition for antitony switching mappings and connection with complemention in sections have been shown.