Ordered semigroup

Ordered semigroup

An ordered semigroup is a set equipped with an operation that satisfies the semigroup axioms and a partial order induced by the operation, where the partial order is defined such that for any two elements in the set, one element is less than or equal to the other if and only if the second element can be obtained by adding an element from the set to the first element.From: Mathematical Aspects of Logic Programming Semantics [2016], Nonlinear Digital Filtering with Python [2019]

Fixed-Point Theory for Generalized Metric Spaces

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Published in Pascal Hitzler, Anthony Seda, Mathematical Aspects of Logic Programming Semantics, 2016

Pascal Hitzler, Anthony Seda

By an ordered semigroup with identity we mean a semigroup A with 0, say, on which there is defined an ordering ≤ satisfying: 0 ≤ a for all a ∈ A, and if a1 ≤ a2 and a’1≤ a’2, then a1 + a’1≤ a1 + a’2 for all a1, a’1, a2, a’2 ∈ A.

Observables on lexicographic MV-algebras

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Published in International Journal of General Systems, 2019

Anatolij Dvurečenskij

Theorems 5.8 and 5.9 allow us to define a partial order, called the Olson order, on the set of observables on an MV-algebra M as well as the sum of any two observables. It is shown that with respect to this order and sum, is a distributive lattice and a lattice-ordered semigroup, Theorem 6.7.

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Fixed-Point Theory for Generalized Metric Spaces

Observables on lexicographic MV-algebras