Transitive Closures of Binary Relations Ii

Transitive closures of the covering relation in semilattices are investigated. Vyšetřuj́ı se tranzitivńı uzávěry pokrývaćı relace v polosvazech. This very short note is an immediate continuation of [1]. We therefore refer to [1] as for terminology, notation, various remarks, further references, etc. 1. The covering relation in semilattices Throughout the note, let S = S(+) be a semilattice (i. e., a commutative idempotent semigroup). Define a relation α on S by (a, b) ∈ α if and only if a+ b = b. 1.1. Proposition. (i) The relation α is a stable (reflexive) ordering of the semilattice. (ii) (a, a + b) ∈ α and (b, a + b) ∈ α for all a, b ∈ S (in fact, a + b = supα(a, b)). (iii) An element a ∈ S is maximal in S(α) (i. e., a is right α-isolated) if and only if a = oS is an absorbing element of S; then a is the (unique) greatest element of S(α). (iv) An element a ∈ S is minimal in S(α) (i. e., a is left α-isolated) if and only if a / ∈ (S \ {a}) +S (then the set (S \ {a}) +S is a proper ideal of S). (v) An element a ∈ S is the smallest element of S(α) if and only if a = 0S is a neutral element of S. Proof. It is obvious. ¤ 1.2. Lemma. (i) Every weakly pseudoirreducible finite α-sequence is pseudoirreducible. (ii) Every weakly pseudoirreducible right (left, resp.) directed infinite α-sequence is pseudoirreducible. (iii) If there exists no pseudoirreducible right directed infinite α-sequence then oS ∈ S. Proof. It is obvious (combine (ii), 1.1(iii) and I.5.4(iii)). ¤ The work is a part of the research project MSM0021620839 financed by MŠMT and partly supported by the Grant Agency of the Czech Republic, grant #201/05/0002.