An element e of an ordered semigroup <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\left( S,\cdot ,\le \right) $</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mi>S</mi> <mo>,</mo> <mo>·</mo> <mo>,</mo> <mo>≤</mo> </mfenced> </math></EquationSource> </InlineEquation> is called idempotent (resp. generalised idempotent) if <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$e\le {{e}^{2}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>e</mi> <mo>≤</mo> <msup> <mrow> <mi>e</mi> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> (resp. <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\left( e,{{e}^{2}} \right) \in {{\Re }_{\le }}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfenced close=")" open="("> <mi>e</mi> <mo>,</mo> <msup> <mrow> <mi>e</mi> </mrow> <mn>2</mn> </msup> </mfenced> <mo>∈</mo> <msub> <mi>ℜ</mi> <mo>≤</mo> </msub> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">${{\Re }_{\le }}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℜ</mi> <mo>≤</mo> </msub> </math></EquationSource> </InlineEquation> is the smallest congruence on S containing the relation <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\le \cup {{\le }^{-1}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>≤</mo> <mo>∪</mo> <msup> <mrow> <mo>≤</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>). The set of all idempotents (resp. generalised idempotents) of S is denoted by <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$E\left( S \right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>E</mi> <mfenced close=")" open="("> <mi>S</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> (resp. <InlineEquation ID="IEq7"> <EquationSource Format="TEX">${{E}^{G}}\left( S \right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi>E</mi> </mrow> <mi>G</mi> </msup> <mfenced close=")" open="("> <mi>S</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation>). S is called orthodox if (i) the set <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$E\left( S \right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>E</mi> <mfenced close=")" open="("> <mi>S</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> is non empty and (ii) <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$ef\in {{E}^{G}}\left( S \right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>e</mi> <mi>f</mi> <mo>∈</mo> <msup> <mrow> <mi>E</mi> </mrow> <mi>G</mi> </msup> <mfenced close=")" open="("> <mi>S</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> for every <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$e,f\in E\left( S \right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>e</mi> <mo>,</mo> <mi>f</mi> <mo>∈</mo> <mi>E</mi> <mfenced close=")" open="("> <mi>S</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation>. An element x in S is an inverse (resp. generalised inverse) of an element a of S if <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$a\le axa$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>≤</mo> <mi>a</mi> <mi>x</mi> <mi>a</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$x\le xax$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>≤</mo> <mi>x</mi> <mi>a</mi> <mi>x</mi> </mrow> </math></EquationSource> </InlineEquation> (resp. <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\left( a,axa \right) ,\left( x,xax \right) \in {{\Re }_{\le }}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfenced close=")" open="("> <mi>a</mi> <mo>,</mo> <mi>a</mi> <mi>x</mi> <mi>a</mi> </mfenced> <mo>,</mo> <mfenced close=")" open="("> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mi>a</mi> <mi>x</mi> </mfenced> <mo>∈</mo> <msub> <mi>ℜ</mi> <mo>≤</mo> </msub> </mrow> </math></EquationSource> </InlineEquation>). We study the notions of generalised inverse and generalised idempotent element and we show that, in an orthodox ordered semigroup, if we know a single generalised inverse of an element a, then we know the set of all generalised inverses of a. We also study the structure of orthodox ordered semigroups giving basic properties of orthodox ordered semigroups and equivalent conditions (based on inverse, generalised inverse, idempotent, generalised idempotent elements) according to which an ordered semigroup is orthodox.

Orthodox ordered semigroups

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2026年2月2日

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Michael Tsingelis

摘要

An element e of an ordered semigroup $\left( S,\cdot ,\le \right) $ $(S, \cdot, \leq)$ is called idempotent (resp. generalised idempotent) if $e\le {{e}^{2}}$ $e \leq e^{2}$ (resp. $\left( e,{{e}^{2}} \right) \in {{\Re }_{\le }}$ $(e, e^{2}) \in ℜ_{\leq}$ where ${{\Re }_{\le }}$ $ℜ_{\leq}$ is the smallest congruence on S containing the relation $\le \cup {{\le }^{-1}}$ $\leq \cup \leq^{- 1}$ ). The set of all idempotents (resp. generalised idempotents) of S is denoted by $E\left( S \right) $ $E (S)$ (resp. ${{E}^{G}}\left( S \right) $ $E^{G} (S)$ ). S is called orthodox if (i) the set $E\left( S \right) $ $E (S)$ is non empty and (ii) $ef\in {{E}^{G}}\left( S \right) $ $e f \in E^{G} (S)$ for every $e,f\in E\left( S \right) $ $e, f \in E (S)$ . An element x in S is an inverse (resp. generalised inverse) of an element a of S if $a\le axa$ $a \leq a x a$ and $x\le xax$ $x \leq x a x$ (resp. $\left( a,axa \right) ,\left( x,xax \right) \in {{\Re }_{\le }}$ $(a, a x a), (x, x a x) \in ℜ_{\leq}$ ). We study the notions of generalised inverse and generalised idempotent element and we show that, in an orthodox ordered semigroup, if we know a single generalised inverse of an element a, then we know the set of all generalised inverses of a. We also study the structure of orthodox ordered semigroups giving basic properties of orthodox ordered semigroups and equivalent conditions (based on inverse, generalised inverse, idempotent, generalised idempotent elements) according to which an ordered semigroup is orthodox.