Comparing Numbers of Diagonal Subsemigroups and Congruences for Semigroups
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| Format: | Preprint |
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2026
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| _version_ | 1866910026926391296 |
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| author | Barber, Callum Ruškuc, Nik |
| author_facet | Barber, Callum Ruškuc, Nik |
| contents | Given a semigroup $S$, a diagonal subsemigroup $ρ$ is defined to be a reflexive and compatible relation on $S$, i.e. a subsemigroup of the direct square $S\times S$ containing the diagonal $\{ (s,s)\colon s\in S\}$. When $S$ is finite, we define the DSC coefficient $χ(S)$ to be the ratio of the number of congruences to the number of diagonal subsemigroups. In a previous work we observed that $χ(S) = 1$ if and only if $S$ is a group. Here we show that for any rational $α$ with $0 < α\leq 1$, there exists a semigroup with $χ(S) = α$. We do this by utilizing the Rees matrix construction and adapting the congruence classification of such semigroups to describe their diagonal subsemigroups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_17286 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Comparing Numbers of Diagonal Subsemigroups and Congruences for Semigroups Barber, Callum Ruškuc, Nik Rings and Algebras Combinatorics Group Theory 08A30, 20M10, 05E16 Given a semigroup $S$, a diagonal subsemigroup $ρ$ is defined to be a reflexive and compatible relation on $S$, i.e. a subsemigroup of the direct square $S\times S$ containing the diagonal $\{ (s,s)\colon s\in S\}$. When $S$ is finite, we define the DSC coefficient $χ(S)$ to be the ratio of the number of congruences to the number of diagonal subsemigroups. In a previous work we observed that $χ(S) = 1$ if and only if $S$ is a group. Here we show that for any rational $α$ with $0 < α\leq 1$, there exists a semigroup with $χ(S) = α$. We do this by utilizing the Rees matrix construction and adapting the congruence classification of such semigroups to describe their diagonal subsemigroups. |
| title | Comparing Numbers of Diagonal Subsemigroups and Congruences for Semigroups |
| topic | Rings and Algebras Combinatorics Group Theory 08A30, 20M10, 05E16 |
| url | https://arxiv.org/abs/2602.17286 |