The full transformation semigroups <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {T}_{n} $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">T</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$n\in {\mathbb {N}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>, consisting of all maps from a set of cardinality n to itself, are arguably the most important family of finite semigroups. This article investigates the endomorphism monoid <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\textrm{End}(\mathcal {T}_{n}) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>End</mtext> <mo stretchy="false">(</mo> <msub> <mi mathvariant="script">T</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {T}_{n} $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">T</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>. The determination of the elements of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\textrm{End}(\mathcal {T}_{n}) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>End</mtext> <mo stretchy="false">(</mo> <msub> <mi mathvariant="script">T</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is due Schein and Teclezghi. Surprisingly, the algebraic structure of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\textrm{End}(\mathcal {T}_{n}) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>End</mtext> <mo stretchy="false">(</mo> <msub> <mi mathvariant="script">T</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> has not been further explored. We describe Green’s relations and extended Green’s relations on <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\textrm{End}(\mathcal {T}_{n}) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>End</mtext> <mo stretchy="false">(</mo> <msub> <mi mathvariant="script">T</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, and the generalised regularity properties of these monoids. In particular, we prove that <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\mathop {\mathscr {H} } =\mathop {\mathscr {L} } \subseteq \mathop {\mathscr {R} } =\mathop {\mathscr {D} } =\mathop {\mathscr {J} } $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">H</mi> <mo>=</mo> <mi mathvariant="script">L</mi> <mo>⊆</mo> <mi mathvariant="script">R</mi> <mo>=</mo> <mi mathvariant="script">D</mi> <mo>=</mo> <mi mathvariant="script">J</mi> </mrow> </math></EquationSource> </InlineEquation> (with equality if and only if <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$n=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>); the idempotents of <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\textrm{End}(\mathcal {T}_{n}) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>End</mtext> <mo stretchy="false">(</mo> <msub> <mi mathvariant="script">T</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> form a band (which is equal to <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\textrm{End}(\mathcal {T}_{n}) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>End</mtext> <mo stretchy="false">(</mo> <msub> <mi mathvariant="script">T</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> if and only if <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$n=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>) and also the regular elements of <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\textrm{End}(\mathcal {T}_{n}) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>End</mtext> <mo stretchy="false">(</mo> <msub> <mi mathvariant="script">T</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> form a subsemigroup (which is equal to <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$\textrm{End}(\mathcal {T}_{n}) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>End</mtext> <mo stretchy="false">(</mo> <msub> <mi mathvariant="script">T</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> if and only if <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$n\le 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≤</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>). Further, the regular elements of <InlineEquation ID="IEq19"> <EquationSource Format="TEX">$\textrm{End}(\mathcal {T}_{n}) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>End</mtext> <mo stretchy="false">(</mo> <msub> <mi mathvariant="script">T</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> are precisely the idempotents together with all endomorphisms of rank greater than 3. We also provide a presentation for <InlineEquation ID="IEq20"> <EquationSource Format="TEX">$\textrm{End}(\mathcal {T}_{n}) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>End</mtext> <mo stretchy="false">(</mo> <msub> <mi mathvariant="script">T</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with respect to a minimal generating set.

The structure of End($\mathcal {T}_{n} $)

期刊论文
发表日期：

2025年12月12日

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Victoria Gould,
Ambroise Grau,
Marianne Johnson

摘要

The full transformation semigroups $\mathcal {T}_{n} $ $T_{n}$ , where $n\in {\mathbb {N}}$ $n \in N$ , consisting of all maps from a set of cardinality n to itself, are arguably the most important family of finite semigroups. This article investigates the endomorphism monoid $\textrm{End}(\mathcal {T}_{n}) $ $End (T_{n})$ of $\mathcal {T}_{n} $ $T_{n}$ . The determination of the elements of $\textrm{End}(\mathcal {T}_{n}) $ $End (T_{n})$ is due Schein and Teclezghi. Surprisingly, the algebraic structure of $\textrm{End}(\mathcal {T}_{n}) $ $End (T_{n})$ has not been further explored. We describe Green’s relations and extended Green’s relations on $\textrm{End}(\mathcal {T}_{n}) $ $End (T_{n})$ , and the generalised regularity properties of these monoids. In particular, we prove that $\mathop {\mathscr {H} } =\mathop {\mathscr {L} } \subseteq \mathop {\mathscr {R} } =\mathop {\mathscr {D} } =\mathop {\mathscr {J} } $ $H = L \subseteq R = D = J$ (with equality if and only if $n=1$ $n = 1$ ); the idempotents of $\textrm{End}(\mathcal {T}_{n}) $ $End (T_{n})$ form a band (which is equal to $\textrm{End}(\mathcal {T}_{n}) $ $End (T_{n})$ if and only if $n=1$ $n = 1$ ) and also the regular elements of $\textrm{End}(\mathcal {T}_{n}) $ $End (T_{n})$ form a subsemigroup (which is equal to $\textrm{End}(\mathcal {T}_{n}) $ $End (T_{n})$ if and only if $n\le 2$ $n \leq 2$ ). Further, the regular elements of $\textrm{End}(\mathcal {T}_{n}) $ $End (T_{n})$ are precisely the idempotents together with all endomorphisms of rank greater than 3. We also provide a presentation for $\textrm{End}(\mathcal {T}_{n}) $ $End (T_{n})$ with respect to a minimal generating set.

The structure of End(\(\mathcal {T}_{n} \))

摘要