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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2412.14281 |
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| _version_ | 1866915404911214592 |
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| author | Glasscock, Daniel Hindman, Neil Strauss, Dona |
| author_facet | Glasscock, Daniel Hindman, Neil Strauss, Dona |
| contents | In any semigroup $S$ satisfying the Strong Folner Condition, there are three natural notions of density for a subset $A$ of $S$: Folner density $d(A)$, Banach density $d^*(A)$, and translation density $d_t(A)$. If $S$ is commutative or left cancellative, it is known that these three notions coincide. We shall show that these notions coincide for every semigroup $S$ which satisfies the Strong Folner Condition. Using this fact, we solve a problem that has been open for decades, showing that the set of ultrafilters every member of which has positive Folner density is a two sided ideal of $βS$. We also show that, if $S$ is a left amenable semigroup, then the set of ultrafilters every member of which has positive Banach density is a two sided ideal of $βS$. We investigate the density properties of subsets of $S$ in the case in which the minimal left ideals of the Stone-Čech compactification $βS$ are singletons. This occurs in many familiar examples, including all semilattices and all semigroups which have a right zero. We show that this is equivalent to the statement that $S$ satisfies the Strong Folner Condition and that, for every subset $A$ of $S$, $d(A)\in \{0,1\}$. We also examine the relation between the density properties of two semigroups when one is a quotient of the other. The Folner density of a subset of $S$ is always determined by some Folner net in $S$. We show that an arbitrary Folner net in $S$ determines the density of all of the subsets of $S$. And we prove that, if $S$ and $T$ are left amenable semigroups, then $d^*(A\times B)=d^*(A)d^*(B)$ for every subset $A$ of $S$ and every subset $B$ of $T$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_14281 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Følner, Banach, and translation density are equal and other new results about density in left amenable semigroups Glasscock, Daniel Hindman, Neil Strauss, Dona Combinatorics 20M10, 05C42 In any semigroup $S$ satisfying the Strong Folner Condition, there are three natural notions of density for a subset $A$ of $S$: Folner density $d(A)$, Banach density $d^*(A)$, and translation density $d_t(A)$. If $S$ is commutative or left cancellative, it is known that these three notions coincide. We shall show that these notions coincide for every semigroup $S$ which satisfies the Strong Folner Condition. Using this fact, we solve a problem that has been open for decades, showing that the set of ultrafilters every member of which has positive Folner density is a two sided ideal of $βS$. We also show that, if $S$ is a left amenable semigroup, then the set of ultrafilters every member of which has positive Banach density is a two sided ideal of $βS$. We investigate the density properties of subsets of $S$ in the case in which the minimal left ideals of the Stone-Čech compactification $βS$ are singletons. This occurs in many familiar examples, including all semilattices and all semigroups which have a right zero. We show that this is equivalent to the statement that $S$ satisfies the Strong Folner Condition and that, for every subset $A$ of $S$, $d(A)\in \{0,1\}$. We also examine the relation between the density properties of two semigroups when one is a quotient of the other. The Folner density of a subset of $S$ is always determined by some Folner net in $S$. We show that an arbitrary Folner net in $S$ determines the density of all of the subsets of $S$. And we prove that, if $S$ and $T$ are left amenable semigroups, then $d^*(A\times B)=d^*(A)d^*(B)$ for every subset $A$ of $S$ and every subset $B$ of $T$. |
| title | Følner, Banach, and translation density are equal and other new results about density in left amenable semigroups |
| topic | Combinatorics 20M10, 05C42 |
| url | https://arxiv.org/abs/2412.14281 |