2023/03/07 by Sarkar, Mosarof, Singh, Shubh N.
#15A03 #15A04 #20M17 #20M20 #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2303.03861
Let T(X) (resp. L(V)) be the semigroup of all transformations (resp. linear transformations) of a set X (resp. vector space V). For a subset Y of X and a subsemigroup \mathbbS(Y) of T(Y), consider the subsemigroup T_\mathbbS(Y)(X) = \f∈ T(X)\colon f\upharpoonrightY ∈ \mathbbS(Y)\ of T(X), where f\upharpoonrightY∈ T(Y) agrees with f on Y. We give a new characterization for T_\mathbbS(Y)(X) to be a regular semigroup [inverse semigroup]. For a subspace W of V and a subsemigroup \mathbbS(W) of L(W), we define an analogous subsemigroup L_\mathbbS(W)(V) = \f∈ L(V) \colon f\upharpoonrightW ∈ \mathbbS(W)\ of L(V). We describe regular elements in L_\mathbbS(W)(V) and determine when L_\mathbbS(W)(V) is a regular semigroup [inverse semigroup, completely regular semigroup]. If \mathbbS(Y) (resp. \mathbbS(W)) contains the identity of T(Y) (resp. L(W)), we describe unit-regular elements in T_\mathbbS(Y)(X) (resp. L_\mathbbS(W)(V)) and determine when T_\mathbbS(Y)(X) (resp. L_\mathbbS(W)(V)) is a unit-regular semigroup.