arXiv subjects

Ying-Ying Feng

Publications and source records attributed to Ying-Ying Feng.

4 recordsLinked to original sources

arXiv2020

Relationships among quasivarieties induced by the min networks on inverse semigroups

A congruence on an inverse semigroup $S$ is determined uniquely by its kernel and trace. Denoting by $ρ_k$ and $ρ_t$ the least congruence on $S$ having the same kernel and the same trace as $ρ$, respectively, and denoting by $ω$ the universal congruence on $S$, we consider the sequence $ω$, $ω_k$, $ω_t$, $(ω_k)_t$, $(ω_t)_k$, $((ω_k)_t)_k$, $((ω_t)_k)_t$, $\cdots$. The quotients $\{S/ω_k\}$, $\{S/ω_t\}$, $\{S/(ω_k)_t\}$, $\{S/(ω_t)_k\}$, $\{S/((ω_k)_t)_k\}$, $\{S/((ω_t)_k)_t\}$, $\cdots$, as $S$ runs over all inverse semigroups, form quasivarieties. This article explores the relationships among these quasivarieties.

math.GR↗

arXiv2019

A new approach to a network of congruences on an inverse semigroup

This paper enriches the list of known properties of congruence sequences starting from the universal relation and successively performing the operators lower $k$ and lower $t$. Two series of inverse semigroups, namely $\ker{α_n}$-is-Clifford semigroups and $β_n$-is-over-$E$-unitary semigroups, are investigated. Two congruences, namely $α_{n+2}$ and $β_{n+2}$, are found to be the least $\ker{α_n}$-is-Clifford and least $β_n$-is-over-$E$-unitary congruences on $S$, respectively. A new system of implications is established for the quasivarieties of inverse semigroups induced by the min network.

math.GR↗

arXiv2019

Some special congruences on completely regular semigroups

This paper enriches the list of properties of the congruence sequences starting from the universal relation and successively performing the operations of lower $t$ and lower $k$. Three classes of completely regular semigroups, namely semigroups for which $\kerσ$ is a cryptogroup, semigroups for which $\kerν$ is a cryptogroup and semigroups for which $κ$ is over rectangular bands, are studied. $((ω_t)_k)_t$, $((\mathcal{D}_t)_k)_t$ and $((ω_k)_t)_k$ are found to be the least congruences on $S$ such that the quotient semigroups are semigroups for which $\kerσ$ is a cryptogroup, $\kerν$ is a cryptogroup and $κ$ is over rectangular bands, respectively. The results obtained present a response to three problems in Petrich and Reilly's textbook \textquoteleft\textquoteleft Completely Regular Semigroups\textquoteright\textquoteright.

math.GR↗

arXiv2018

Presentations for singular wreath products

For a monoid $M$ and a subsemigroup $S$ of the full transformation semigroup $T_n$, the wreath product $M\wr S$ is defined to be the semidirect product $M^n\rtimes S$, with the coordinatewise action of $S$ on $M^n$. The full wreath product $M\wr T_n$ is isomorphic to the endomorphism monoid of the free $M$-act on $n$ generators. Here, we are particularly interested in the case that $S=Sing_n$ is the singular part of $T_n$, consisting of all non-invertible transformations. Our main results are presentations for $M\wr Sing_n$ in terms of certain natural generating sets, and we prove these via general results on semidirect products and wreath products. We re-prove a classical result of Bulman-Fleming that $M\wr Sing_n$ is idempotent generated if and only if the set $M/L$ of $L$-classes of $M$ forms a chain under the usual ordering of $L$-classes, and we give a presentation for $M\wr Sing_n$ in terms of idempotent generators for such a monoid $M$. Among other results, we also give estimates for the minimal size of a generating set for $M\wr Sing_n$, as well as exact values in some cases (including the case that $M$ is finite and $M/L$ is a chain, in which case we also calculate the minimal size of an idempotent generating set). As an application of our results, we obtain a presentation (with idempotent generators) for the idempotent generated subsemigroup of the endomorphism monoid of a uniform partition of a finite set.

math.GR↗