Isotopes of nearlattices

This thesis studies the nature of isotopes of a nearlattice.A nearlattice is a lower semilattice with the upper bcund property, which says that any two elements possess a supremum whenever they have a common upper bound.The topic arose out of a study on the kernels, around a particular element n , of a skeletal congruence on a distributive lattice.Also, we found that the idea of an isotope was very fruitful in extending results on ideals of nearlattices to the w-ideals; that is, convex subnearlattices containing n .Chapter 1 discusses ideals, join-partial congruences and other results on nearlattices which are basic to this thesis.Chapters 2 and 3 introduce the notions of standard element, neutral element and central element of a nearlattice.These concepts are essential for the further developments of Chapters h and 5. Also in Chapter 3, we discuss direct summands and multipliers of a nearlattice and generalize a number of results of [6] and [9].Chapter h introduces the concept of an isotope.Two new types of elements arise; one is called superstandard and the other is nearlyexists for all x, y Z S , while the nearlattice S is medial if each of its elements is medial.A sesquimedial element is a strengthening of a medial element, but in a medial nearlattice every element is sesquimedial.For a medial element n of a nearlattice S , we can form a new binary operation