A Study on Generalized Lattices

Abstract

The concept of partial ordering plays an important role in the study newlineof mathematical theories. Ideas of supremum and infimum and the development newlineof lattice theory is a milestone in theory of partial ordering. newlineTransitive property is an important requirement for lattices as it is necessary newlinefor the associativity property of the operations of supremum and newlineinfimum. E.Fried introduced weakly associative lattices as a generalization newlineof latices by deleting the axiom of associativity from the set of axioms newlinefor lattices. In this thesis we have studied the properties of these weakly newlineassociative lattices. newlineConcept of nodes in trellises is introduced as a generalization of nodes in newlinelattices and it is proved that set of all nodes of a trellis forms a distributive newlinelattice. We have proved that the node of a trellis is standard if and only newlineif it is not contained in a non trivial cycle. Congruence relations in direct newlineproduct of trellises is introduced and it is proved that a binary relation newlineand#936; on the direct product of trellises L and K is a congruence relation if and only if it is of the form and#920;×and#934; where and#920; is a congruence relation on L newlineand and#934; is a congruence relation on K. The notions of modular and weakly newlinedistributive trellises are introduced and the relation between distributivity, newlinemodularity, and weak distributivity is obtained. newline