On Proximal Relations in Transformation Semigroups Arising from Generalized Shifts

For a finite discrete topological space X with at least two elements, a nonempty set F, and a map phi : Gamma -> Gamma, sigma(phi), : X-Gamma -> X-Gamma with sigma(phi)((x(alpha))(alpha subset of Gamma)) = (x(phi)(alpha))alpha subset of Gamma (for (x,),(alpha is an element of Gamma) is an element of X-Gamma) is a generalized shift. In this text for S = {sigma(psi) : psi is an element of Gamma(Gamma)} and = {crip : F F is bijective} we study proximal relations of transformation semigroups (S, X-Gamma) and (1-1, X-Gamma). Regarding proximal relation we prove: P (S, X1) = {((x,),(alpha subset of Gamma), (y,)(alpha subset of Gamma)) (alpha subset of Gamma) x Gamma(Gamma) : J3 E F (Gamma(Gamma) = Gamma(Gamma))} and P(H, X-Gamma) C {((x0,1, 0, (yo,)(alpha subset of Gamma)) E Xr x Xr : {p E F : xo = yp} is infinite} U {(x,x) : x is an element of X-Gamma}. Moreover, for infinite F, both transformation semigroups (S, X-Gamma) and (H1, X-Gamma) are regionally proximal, i.e., Q(S, = = Xr x XI', also for sydetically proximal relation we have LP-1, = {((x,)(alpha subset of Gamma) (y(alpha))(alpha subset of Gamma) is an element of X-Gamma x X-Gamma : {gamma is an element of Gamma : x gamma not equal y gamma}is finite}.