The semigroup of ultrafilters near zero as the adequacy kernel of a partial semigroup

Document Type : Original Article

Authors

1 Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Guilan, Iran

2 Department of Mathematics, University of Qom, Qom, Iran

Abstract

‎Let $T$ be a dense subsemigroup of $((0,\infty),+)$ and put $S = T \cap (0,1)$‎. ‎In this paper we show that $(S,\dot{+})$ is a commutative adequate partial ‎semigroup‎, ‎where $x \dot{+} y = x+y$ whenever $x+y \in S$‎. ‎First‎, ‎we prove that the semigroup of ultrafilters near zero of $S$ is an‎ ‎adequacy kernel of $S$; consequently‎, ‎every result that holds for ‎commutative adequate partial semigroups also holds for the semigroup of‎ ‎ultrafilters near zero‎. ‎Second‎, ‎we demonstrate that $(S,\dot{+})$ does not satisfy the $(SFC)$‎ ‎condition for commutative adequate partial semigroups‎. ‎Finally‎, ‎we propose a F{\o}lner density equipped with a suitable operation on $S$‎, ‎which aids the study of density near zero‎.

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References

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Measure Algebras and Applications
Volume 3, Issue 2 - Serial Number 5
December 2025
Pages 1-11

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