<p>The concept of ideals in algebraic structures plays an important role in studying their structure. In this paper, we enrich an algebraic structure, which is a generalization of ordered groups called ordered semigroups, by using the more general form of fuzzy subsemigroups and fuzzy (generalized) bi-ideals. In this aim, the idea of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(\overline{\in },\overline{\in }\vee \overline{(\textrm{k}^,\textrm{q}_{\textrm{k}})})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mover> <mo>∈</mo> <mo>¯</mo> </mover> <mo>,</mo> <mover> <mo>∈</mo> <mo>¯</mo> </mover> <mo>∨</mo> <mover> <mrow> <mo stretchy="false">(</mo> <msup> <mtext>k</mtext> <mo>∗</mo> </msup> <mo>,</mo> <msub> <mtext>q</mtext> <mtext>k</mtext> </msub> <mo stretchy="false">)</mo> </mrow> <mo>¯</mo> </mover> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-fuzzy subsemigroups in ordered semigroups is firstly defined. Then, we show that every fuzzy subsemigroup is a <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(\overline{\in },\overline{\in }\vee \overline{(\textrm{k}^,\textrm{q}_{\textrm{k}})})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mover> <mo>∈</mo> <mo>¯</mo> </mover> <mo>,</mo> <mover> <mo>∈</mo> <mo>¯</mo> </mover> <mo>∨</mo> <mover> <mrow> <mo stretchy="false">(</mo> <msup> <mtext>k</mtext> <mo>∗</mo> </msup> <mo>,</mo> <msub> <mtext>q</mtext> <mtext>k</mtext> </msub> <mo stretchy="false">)</mo> </mrow> <mo>¯</mo> </mover> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-fuzzy subsemigroup but the converse is not true in general. An equivalent condition for a fuzzy set to be a <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(\overline{\in },\overline{\in }\vee \overline{(\textrm{k}^,\textrm{q}_{\textrm{k}})})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mover> <mo>∈</mo> <mo>¯</mo> </mover> <mo>,</mo> <mover> <mo>∈</mo> <mo>¯</mo> </mover> <mo>∨</mo> <mover> <mrow> <mo stretchy="false">(</mo> <msup> <mtext>k</mtext> <mo>∗</mo> </msup> <mo>,</mo> <msub> <mtext>q</mtext> <mtext>k</mtext> </msub> <mo stretchy="false">)</mo> </mrow> <mo>¯</mo> </mover> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-fuzzy subsemigroup is provided. Additionally, the notions of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(\overline{\in },\overline{\in }\vee \overline{(\textrm{k}^,\textrm{q}_{\textrm{k}})})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mover> <mo>∈</mo> <mo>¯</mo> </mover> <mo>,</mo> <mover> <mo>∈</mo> <mo>¯</mo> </mover> <mo>∨</mo> <mover> <mrow> <mo stretchy="false">(</mo> <msup> <mtext>k</mtext> <mo>∗</mo> </msup> <mo>,</mo> <msub> <mtext>q</mtext> <mtext>k</mtext> </msub> <mo stretchy="false">)</mo> </mrow> <mo>¯</mo> </mover> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-fuzzy generalized bi-ideals and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(\overline{\in },\overline{\in }\vee \overline{(\textrm{k}^,\textrm{q}_{\textrm{k}})})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mover> <mo>∈</mo> <mo>¯</mo> </mover> <mo>,</mo> <mover> <mo>∈</mo> <mo>¯</mo> </mover> <mo>∨</mo> <mover> <mrow> <mo stretchy="false">(</mo> <msup> <mtext>k</mtext> <mo>∗</mo> </msup> <mo>,</mo> <msub> <mtext>q</mtext> <mtext>k</mtext> </msub> <mo stretchy="false">)</mo> </mrow> <mo>¯</mo> </mover> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-fuzzy bi-ideals in ordered semigroups are also defined. Several conditions are given for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$(\overline{\in },\overline{\in }\vee \overline{(\textrm{k}^, \textrm{q}_{\textrm{k}})})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mover> <mo>∈</mo> <mo>¯</mo> </mover> <mo>,</mo> <mover> <mo>∈</mo> <mo>¯</mo> </mover> <mo>∨</mo> <mover> <mrow> <mo stretchy="false">(</mo> <msup> <mtext>k</mtext> <mo>∗</mo> </msup> <mo>,</mo> <msub> <mtext>q</mtext> <mtext>k</mtext> </msub> <mo stretchy="false">)</mo> </mrow> <mo>¯</mo> </mover> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-fuzzy generalized bi-ideals to be <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$(\overline{\in },\overline{\in }\vee \overline{(\textrm{k}^,\textrm{q}_{\textrm{k}})})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mover> <mo>∈</mo> <mo>¯</mo> </mover> <mo>,</mo> <mover> <mo>∈</mo> <mo>¯</mo> </mover> <mo>∨</mo> <mover> <mrow> <mo stretchy="false">(</mo> <msup> <mtext>k</mtext> <mo>∗</mo> </msup> <mo>,</mo> <msub> <mtext>q</mtext> <mtext>k</mtext> </msub> <mo stretchy="false">)</mo> </mrow> <mo>¯</mo> </mover> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-fuzzy bi-ideals. We prove that each fuzzy (generalized) bi-ideal is <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$(\overline{\in },\overline{\in }\vee \overline{(\textrm{k}^,\textrm{q}_{\textrm{k}})})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mover> <mo>∈</mo> <mo>¯</mo> </mover> <mo>,</mo> <mover> <mo>∈</mo> <mo>¯</mo> </mover> <mo>∨</mo> <mover> <mrow> <mo stretchy="false">(</mo> <msup> <mtext>k</mtext> <mo>∗</mo> </msup> <mo>,</mo> <msub> <mtext>q</mtext> <mtext>k</mtext> </msub> <mo stretchy="false">)</mo> </mrow> <mo>¯</mo> </mover> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-fuzzy (generalized) bi-ideal; however, the converse is not true, as demonstrated by an example. Furthermore, we provide an equivalent condition for the <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$(\overline{\in },\overline{\in }\vee \overline{(\textrm{k}^*,\textrm{q}_{\textrm{k}})})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mover> <mo>∈</mo> <mo>¯</mo> </mover> <mo>,</mo> <mover> <mo>∈</mo> <mo>¯</mo> </mover> <mo>∨</mo> <mover> <mrow> <mo stretchy="false">(</mo> <msup> <mtext>k</mtext> <mo>∗</mo> </msup> <mo>,</mo> <msub> <mtext>q</mtext> <mtext>k</mtext> </msub> <mo stretchy="false">)</mo> </mrow> <mo>¯</mo> </mover> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-fuzzy (generalized) bi-ideal. Moreover, we characterize <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$(\overline{\in },\overline{\in }\vee \overline{(\textrm{k}^\star ,\textrm{q}_{\textrm{k}})})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mover> <mo>∈</mo> <mo>¯</mo> </mover> <mo>,</mo> <mover> <mo>∈</mo> <mo>¯</mo> </mover> <mo>∨</mo> <mover> <mrow> <mo stretchy="false">(</mo> <msup> <mtext>k</mtext> <mo>⋆</mo> </msup> <mo>,</mo> <msub> <mtext>q</mtext> <mtext>k</mtext> </msub> <mo stretchy="false">)</mo> </mrow> <mo>¯</mo> </mover> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-fuzzy (generalized) bi-ideals using level subsets, <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$(\in \vee (\textrm{k}^\star ,\textrm{q}_{\textrm{k}}))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mo>∈</mo> <mo>∨</mo> <mrow> <mo stretchy="false">(</mo> <msup> <mtext>k</mtext> <mo>⋆</mo> </msup> <mo>,</mo> <msub> <mtext>q</mtext> <mtext>k</mtext> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-level subsets, and characteristic functions. Finally, by applying the ideal of the <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$(\textrm{k}^\star , \textrm{k})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msup> <mtext>k</mtext> <mo>⋆</mo> </msup> <mo>,</mo> <mtext>k</mtext> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-upper part of fuzzy sets, more comprehensive characterizations of <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$(\overline{\in },\overline{\in }\vee \overline{(\textrm{k}^\star ,\textrm{q}_{\textrm{k}})})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mover> <mo>∈</mo> <mo>¯</mo> </mover> <mo>,</mo> <mover> <mo>∈</mo> <mo>¯</mo> </mover> <mo>∨</mo> <mover> <mrow> <mo stretchy="false">(</mo> <msup> <mtext>k</mtext> <mo>⋆</mo> </msup> <mo>,</mo> <msub> <mtext>q</mtext> <mtext>k</mtext> </msub> <mo stretchy="false">)</mo> </mrow> <mo>¯</mo> </mover> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-fuzzy (generalized) bi-ideals are studied.</p>

Extension and structural development of fuzzy (generalized) bi-ideals in ordered semigroups

期刊论文
发表日期：

2026年3月31日

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Ahsan Mahboob,
Ghulam Muhiuddin,
Young Bae Jun,
Bijan Davvaz

摘要

The concept of ideals in algebraic structures plays an important role in studying their structure. In this paper, we enrich an algebraic structure, which is a generalization of ordered groups called ordered semigroups, by using the more general form of fuzzy subsemigroups and fuzzy (generalized) bi-ideals. In this aim, the idea of $(\overline{\in },\overline{\in }\vee \overline{(\textrm{k}^*,\textrm{q}_{\textrm{k}})})$ $(\bar{\in}, \bar{\in} \lor \bar{(k^{*}, q_{k})})$ -fuzzy subsemigroups in ordered semigroups is firstly defined. Then, we show that every fuzzy subsemigroup is a $(\overline{\in },\overline{\in }\vee \overline{(\textrm{k}^*,\textrm{q}_{\textrm{k}})})$ $(\bar{\in}, \bar{\in} \lor \bar{(k^{*}, q_{k})})$ -fuzzy subsemigroup but the converse is not true in general. An equivalent condition for a fuzzy set to be a $(\overline{\in },\overline{\in }\vee \overline{(\textrm{k}^*,\textrm{q}_{\textrm{k}})})$ $(\bar{\in}, \bar{\in} \lor \bar{(k^{*}, q_{k})})$ -fuzzy subsemigroup is provided. Additionally, the notions of $(\overline{\in },\overline{\in }\vee \overline{(\textrm{k}^*,\textrm{q}_{\textrm{k}})})$ $(\bar{\in}, \bar{\in} \lor \bar{(k^{*}, q_{k})})$ -fuzzy generalized bi-ideals and $(\overline{\in },\overline{\in }\vee \overline{(\textrm{k}^*,\textrm{q}_{\textrm{k}})})$ $(\bar{\in}, \bar{\in} \lor \bar{(k^{*}, q_{k})})$ -fuzzy bi-ideals in ordered semigroups are also defined. Several conditions are given for $(\overline{\in },\overline{\in }\vee \overline{(\textrm{k}^*, \textrm{q}_{\textrm{k}})})$ $(\bar{\in}, \bar{\in} \lor \bar{(k^{*}, q_{k})})$ -fuzzy generalized bi-ideals to be $(\overline{\in },\overline{\in }\vee \overline{(\textrm{k}^*,\textrm{q}_{\textrm{k}})})$ $(\bar{\in}, \bar{\in} \lor \bar{(k^{*}, q_{k})})$ -fuzzy bi-ideals. We prove that each fuzzy (generalized) bi-ideal is $(\overline{\in },\overline{\in }\vee \overline{(\textrm{k}^*,\textrm{q}_{\textrm{k}})})$ $(\bar{\in}, \bar{\in} \lor \bar{(k^{*}, q_{k})})$ -fuzzy (generalized) bi-ideal; however, the converse is not true, as demonstrated by an example. Furthermore, we provide an equivalent condition for the $(\overline{\in },\overline{\in }\vee \overline{(\textrm{k}^*,\textrm{q}_{\textrm{k}})})$ $(\bar{\in}, \bar{\in} \lor \bar{(k^{*}, q_{k})})$ -fuzzy (generalized) bi-ideal. Moreover, we characterize $(\overline{\in },\overline{\in }\vee \overline{(\textrm{k}^\star ,\textrm{q}_{\textrm{k}})})$ $(\bar{\in}, \bar{\in} \lor \bar{(k^{⋆}, q_{k})})$ -fuzzy (generalized) bi-ideals using level subsets, $(\in \vee (\textrm{k}^\star ,\textrm{q}_{\textrm{k}}))$ $(\in \lor (k^{⋆}, q_{k}))$ -level subsets, and characteristic functions. Finally, by applying the ideal of the $(\textrm{k}^\star , \textrm{k})$ $(k^{⋆}, k)$ -upper part of fuzzy sets, more comprehensive characterizations of $(\overline{\in },\overline{\in }\vee \overline{(\textrm{k}^\star ,\textrm{q}_{\textrm{k}})})$ $(\bar{\in}, \bar{\in} \lor \bar{(k^{⋆}, q_{k})})$ -fuzzy (generalized) bi-ideals are studied.