Ordered monoid

Ordered monoid

An ordered monoid is a monoid equipped with a partial-order relation "≤" that is compatible with addition, meaning that for any elements a, b, and c in the monoid, if a ≤ b, then a + c ≤ b + c. Additionally, an ordered monoid is a complete lattice and the multiplication is distributive over unions. Examples of ordered monoids can be found in Birkhoff [4] or Blyth and Janowitz [5].From: Mathematical Morphology in Image Processing [2018], Ideals of an EMV-semiring [2020]

Mathematical Morphology with Noncommutative Symmetry Groups

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Published in Edward R. Dougherty, Mathematical Morphology in Image Processing, 2018

Jos B.T.M. Roerdink

Remark 3.5. We notice in passing that P(Γ) is a monoid under the multiplication ⊕Γ, that is, a semigroup with unit element {e}. Since P(Γ)is a complete lattice as well, and the multiplication ⊕Γ is distributive over unions (see the next proposition), we have an example here of a so→called complete lattice→ordered monoid or cl→monoid; see Birkhoff [4] or Blyth and Janowitz [5].

Introduction

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Published in John N. Mordeson, Davender S. Malik, Fuzzy Automata and Languages, 2002

John N. Mordeson, Davender S. Malik

(2) FA is closed under the relative product if A is complete. FA is then a lattice ordered monoid if A is complete and deterministic.

FUZZY LOGIC

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Published in Kumar S. Ray, Soft Computing and Its Applications, Volume One, 2014

Kumar S. Ray

In algebraic terms, such a t-norm ® makes the real units interval into an ordered monoid, that is into an abelian semigroup with unit element. And this ordered monoid is even integral, that is its unit element is at the same time the universal upper bound of the ordering.

Ideals of an EMV-semiring

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Journal Information

Published in International Journal of General Systems, 2020

R. A. Borzooei, M. Shenavaei, A. Di Nola, O. Zahiri

Let be a monoid. An element is called idempotent if a + a = a. The set of all idempotent elements of M is denoted by . A monoid M is called an ordered monoid if it is equipped with a partial-order relation “≤” which is compatible with “+,” that is, for , implies , for any .

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