Prominent GE-Filters and GE-Morphisms in GE-Algebras

Abstract

The relationship between a transitive GE-algebra and a belligerent GE-algebra (also, between an antisymmetric GE-algebra and a left exchangeable GE-algebra) is displayed. A condition for the trivial GE-filter to be a belligerent GE-filter is provided. The least GE-filter containing a given GE-filter and one element is formed. Conditions under which any set can be turned into a GE-filter are described. The notions of circle dot-GE-algebras and prominent GE-filters are introduced, and their properties are investigated. The relationship between a prominent GE-filter and a GE-filter are considered, and conditions for a GE-filter and the trivial GE-filter to be a prominent GE-filter are given. The conditions under which the upset of an element will be a prominent GE-filter are examined, and the extension property for the prominent GE-filter is established. The notion of GE-morphism is introduced, and the GE-morphism theorem is considered. Conditions for the kernel of a GE-morphism to be a belligerent GE-filter are provided, and the condition that if the kernel of a GE-morphism is a belligerent GE-filter, then the trivial GE-filter becomes a belligerent GE-filter is given. A condition for the kernel of a GE-morphism to be a belligerent GE-filter is discussed, and the vice versa is also considered.