Intervals in partial monounary algebras

Abstract

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On directed convex subsets of partial monounary algebras

“…b) card A' > 2 and there are a, 6 G A' such that a = f(b), {x G A': x G dom/, f(x) = 6} = 0 and either f(a) = a or a ^ dom/. In view of [4], 3.8, we obtain that if x E A' -{a}, then x G doing and </(.r) = f(x) and either g(a) = a or a ^ domg. It is obvious that in both cases ( 1) is valid.…”

“…Suppose that 1(^4,/) = I(-4,g). By virtue of [4], Theorem 3.8, (A J) and (A,g) must have the same partition into connected components. If A' is a connected component of (-4,/), then instead of f\A f or g\A' we write / and g, respectively.…”

“…Let us deal with the following cases: a) card A' ^ 2. Then g on A' can be defined in an arbitrary way, only A' must be a connected component of (A,g) (according to [4], 3.8). In this case also DC(A',/) = DC(A',g).…”

“…It is obvious that in both cases ( 1) is valid. c) (A',f) is isomorphic to some of the partial monounary algebras considered in [4], 2.1-2.7. In these sections of [4] we have described (up to isomorphism) all (A', g) with \(A', f) = 1(^4', g).…”

“…This paper can be considered to be a continuation of [2], [3] and [4]; in these papers analogous problems concerning the system of all convex subsets and the system of all intervals of a partial monounary algebra have been studied.…”