Autometrized Lattice Ordered Monoids
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Date
2025-06
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Addis Ababa University
Abstract
In this dissertation, we introduce autometrized lattice ordered monoids (AL-monoids), a new generalization of dually residuated lattice ordered semigroups (DRl-semigroups). We explore various algebraic properties of AL-monoids and investigate isometries within this structure. A key finding is that the set of invertible elements in an AL-monoid forms an l-group, while the set of complemented elements constitutes a Boolean algebra. We further establish that an AL-monoid A with a unity element is a Boolean algebra if the mapping x 7→ a ∗ x is an isometry for every a ∈ A. The geometric aspects of AL-monoids are examined through
the introduction of metric betweenness and related concepts, including B-linearity, D-linearity, lattice betweenness, segments, and equilateral triangles. Notably, we prove that equilateral triangles cannot exist in AL-monoids, subsume the geometric properties of commutative DRlsemigroups.
Moreover, we define various types of ideals within AL-monoids, such as polar ideals, regular ideals, prime ideals, and annihilators, and elucidate their interconnections. We introduce the value of an element for an ideal I(A) and characterize regular ideals based on this value. Finally, we provide characterizations of polar ideals concerning minimal prime ideals, annihilators, and maximal polar ideals, contributing to a deeper understanding of the structure
and relationships within AL-monoids.
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Keywords
Autometrized Lattice, Ordered Monoids