English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

LetT(X)T\left(X)be the full transformation semigroup on a setXX. For an equivalenceEEonXX, letTE∗(X)={α∈T(X):∀x,y∈X,(x,y)∈E⇔(xα,yα)∈E}.{T}_{{E}^{\ast }}\left(X)=\left\{\alpha \in T\left(X):\forall x,y\in X,\left(x,y)\in E\iff \left(x\alpha ,y\alpha )\in E\right\}.For each nonempty subsetYYofXX, we denote the restriction ofEEtoYYbyEY{E}_{Y}. LetTE∗(X,Y){T}_{{E}^{\ast }}\left(X,Y)be the intersection of the semigroupTE∗(X){T}_{{E}^{\ast }}\left(X)with the semigroup of all transformations with restricted rangeYYunder the condition that∣X/E∣=∣Y/EY∣| X\hspace{-0.1em}\text{/}E| =| Y\hspace{-0.16em}\text{/}\hspace{-0.1em}{E}_{Y}|. Equivalently,TE∗(X,Y)={α∈TE∗(X):Xα⊆Y}{T}_{{E}^{\ast }}\left(X,Y)=\left\{\alpha \in {T}_{{E}^{\ast }}\left(X):X\alpha \subseteq Y\right\}, where∣X/E∣=∣Y/EY∣| X\hspace{-0.1em}\text{/}\hspace{-0.1em}E| =| Y\hspace{-0.16em}\text{/}\hspace{-0.1em}{E}_{Y}|. ThenTE∗(X,Y){T}_{{E}^{\ast }}\left(X,Y)is a subsemigroup ofTE∗(X){T}_{{E}^{\ast }}\left(X). In this paper, we characterize the natural partial order onTE∗(X,Y){T}_{{E}^{\ast }}\left(X,Y). Then we find the elements which are compatible and describe the maximal and minimal elements. We also prove that every element ofTE∗(X,Y){T}_{{E}^{\ast }}\left(X,Y)lies between maximal and minimal elements. Finally, the existence of an upper cover and a lower cover is investigated.

A partial order on transformation semigroups with restricted range that preserve double direction equivalence