On Topological Machines

A topological machine is a topo |ogized quas i -machine of G i n s b u r g and is defined by two cont inuous funct ions f and g, T ~ X x S ~ X , the s ta tet rans i t ion funct ion and the output funct ion, respect ively, where .X is a nonvoid T2-space, the state space, S and T are two topological semigroups , the i n p u t semigroup and the ou tpu t semigroup, respect ively, such tha t f (x, s i s 2 ) f ( f (x, sl) , s2) and g(x, s~s2) -g (x , sl) g( f (x, Sl), s~) for all x ~ X and all s l , s.~ c S. In Sect ion I of this paper some resul ts are obtained toward the s t ruc ture of the o u t p u t funct ion for a few special classes of machines and, in Section 2, some basic concepts and results of the algebraic theory are ex tended to the topological case.