<p>A set of characterizations of ultrafilters belonging to the smallest ideal of Stone-Čech compactification of a discrete semigroup are exhibited using syndetic sets, strongly central sets and very strongly central sets respectively. These lead to represent piecewise syndetic subsets of a semigroup in terms of the sets that contain a broken <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation> set, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {A}\in \{$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">A</mi> <mo>∈</mo> <mo stretchy="false">{</mo> </mrow> </math></EquationSource> </InlineEquation>syndetic, quasi-central, central, strongly central, very strongly central<InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\}$</EquationSource> <EquationSource Format="MATHML"><math> <mo stretchy="false">}</mo> </math></EquationSource> </InlineEquation>. A characterization of sets that contain a broken <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\hbox {IP}^{n}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mtext>IP</mtext> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> set (<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>) is obtained using ultrafilters, and it is showed that a set contains a broken <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\hbox {IP}^{n}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mtext>IP</mtext> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> set if and only if it contains a broken IP set. Without assuming the countability of a semigroup, it is established that piecewise syndetic sets, that is, sets that contain a broken syndetic set (broken IP set) are precisely those sets that force uniform recurrence (force recurrence). In conclusion, all the said results are explored near idempotent of a semitopological semigroup.</p>

Polymorphism of piecewise syndetic and broken IP sets which is continued near idempotent

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2025年9月17日

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Ujjal Kumar Hom,
Manoranjan Singha

摘要

A set of characterizations of ultrafilters belonging to the smallest ideal of Stone-Čech compactification of a discrete semigroup are exhibited using syndetic sets, strongly central sets and very strongly central sets respectively. These lead to represent piecewise syndetic subsets of a semigroup in terms of the sets that contain a broken $\mathcal {A}$ $A$ set, where $\mathcal {A}\in \{$ $A \in {$ syndetic, quasi-central, central, strongly central, very strongly central $\}$ $}$ . A characterization of sets that contain a broken $\hbox {IP}^{n}$ ${IP}^{n}$ set ( $n\in \mathbb {N}$ $n \in N$ ) is obtained using ultrafilters, and it is showed that a set contains a broken $\hbox {IP}^{n}$ ${IP}^{n}$ set if and only if it contains a broken IP set. Without assuming the countability of a semigroup, it is established that piecewise syndetic sets, that is, sets that contain a broken syndetic set (broken IP set) are precisely those sets that force uniform recurrence (force recurrence). In conclusion, all the said results are explored near idempotent of a semitopological semigroup.