For a finite discrete topological space X with at least two elements, a
nonempty set Ξ, and a map Ο:ΞβΞ,
ΟΟβ:XΞβXΞ with
ΟΟβ((xΞ±β)Ξ±βΞβ)=(xΟ(Ξ±)β)Ξ±βΞβ (for (xΞ±β)Ξ±βΞββXΞ) is a generalized shift. In this text for
S={ΟΟβ:ΟβΞΞ} and
H={ΟΟβ:ΞβΟΞ is
bijective} we study proximal relations of transformation semigroups
(S,XΞ) and (H,XΞ). Regarding proximal
relation we prove: P(S,XΞ)={((xΞ±β)Ξ±βΞβ,(yΞ±β)Ξ±βΞβ)βXΞΓXΞ:βΞ²βΞ(xΞ²β=yΞ²β)} and
P(H,XΞ)β{((xΞ±β)Ξ±βΞβ,(yΞ±β)Ξ±βΞβ)βXΞΓXΞ:{Ξ²βΞ:xΞ²β=yΞ²β} is
infinite~}βͺ{(x,x):xβX}. \\ Moreover, for infinite
Ξ, both transformation semigroups (S,XΞ) and
(H,XΞ) are regionally proximal, i.e., Q(S,XΞ)=Q(H,XΞ)=XΞΓXΞ, also for
sydetically proximal relation we have L(H,XΞ)={((xΞ±β)Ξ±βΞβ,(yΞ±β)Ξ±βΞβ)βXΞΓXΞ:{Ξ³βΞ:xΞ³βξ =yΞ³β} is
finite}.Comment: 8 page