Ordered sets, semilattices, distributive lattices and Boolean algebras with homomorphic endomorphism semigroups

Isomorphisms between semigroups of isotone maps

“…The preceding result generalizes Theorem 2 of Schein (1970). Comparing 2.2 with the results of McAlister (1971), we notice that by 2.1 a surjective homomorphism F: S -> T between sufficient semigroups is a n isomorphism if a n d only if for <p G S, F(«p) = 0 <=> <p = 0.…”

“…y f = 0. We note that every subsemigroup S of & P which is sufficient in the sense of Schein (1970) is also sufficient in our sense, but not conversely. Instead of (1), (2) Schein uses the conditions (1') if x, y G P and.y ^ 0, then there exists £ G S such that im £ = (0, x} and y£ = x, (2') if x,y E P and x <^y, then there exists <p G S such that x«p ^ 0 and y<p = 0.…”

“…Of particular interest are those cases where an associated semigroup determines the space up to homeomorphism (see Gluskin and others (1977)). Schein (1970) has demonstrated that some results of this type are order theoretic in nature and thus follow from fairly general theorems on isomorphisms between semigroups of isotone maps. We will prove here (Theorem 2.2) a slight generalization of Schein (1970), Theorem 2, which also covers the main results of Magill (1969) and McAlister (1971).…”

“…Schein (1970) has demonstrated that some results of this type are order theoretic in nature and thus follow from fairly general theorems on isomorphisms between semigroups of isotone maps. We will prove here (Theorem 2.2) a slight generalization of Schein (1970), Theorem 2, which also covers the main results of Magill (1969) and McAlister (1971). The range of a mapping <p is denoted by im <p; if im <p contains exactly two elements, then <p is called 2-valued.…”

“…Since for any poset P with 0 there is a n obvious 1-1 correspondence between the maps <p G & P and the isotone partial maps <p on the poset P -{0} (see Schein (1970), p. 46), 2.2 may be restated as…”