<p>The notion of a self-similar ultragraph <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(G,\mathcal {U},\varphi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mi mathvariant="script">U</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and its <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$C^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-algebra <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {O}_{G,\mathcal {U}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">O</mi> <mrow> <mi>G</mi> <mo>,</mo> <mi mathvariant="script">U</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> were introduced in our recent work, where we proposed inverse semigroup and groupoid models for such <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$C^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-algebras as well. In this paper, we investigate minimality and effectiveness of the groupoid of a self-similar ultragraph <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(G,\mathcal {U},\varphi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mi mathvariant="script">U</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. In particular, we obtain a result for simplicity of the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$C^*$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-algebras <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {O}_{G,\mathcal {U}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">O</mi> <mrow> <mi>G</mi> <mo>,</mo> <mi mathvariant="script">U</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> in a certain case.</p>

Minimality and effectiveness of the groupoid associated to a self-similar ultragraph

期刊论文
发表日期：

2025年12月9日

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Hossein Larki,
Najmeh Rajabzadeh-Hasiri

摘要

The notion of a self-similar ultragraph $(G,\mathcal {U},\varphi )$ $(G, U, φ)$ and its $C^*$ $C^{*}$ -algebra $\mathcal {O}_{G,\mathcal {U}}$ $O_{G, U}$ were introduced in our recent work, where we proposed inverse semigroup and groupoid models for such $C^*$ $C^{*}$ -algebras as well. In this paper, we investigate minimality and effectiveness of the groupoid of a self-similar ultragraph $(G,\mathcal {U},\varphi )$ $(G, U, φ)$ . In particular, we obtain a result for simplicity of the $C^*$ $C^{*}$ -algebras $\mathcal {O}_{G,\mathcal {U}}$ $O_{G, U}$ in a certain case.