Transitive Closures of Binary Relations I

Transitive closures of binary relations and relations α with the property that any two α-sequences connecting two given elements are of the same length are investigated. Vyšetřuj́ı se tranzitivńı uzávěry binárńıch relaćı a relaćı α s vlastnost́ı, že každé dvě α-posloupnosti spojuj́ıćı dané dva prvky maj́ı stejnou délku. The present short note collects a few elementary observations concerning the transitive closures of binary relations. All the formulated results are fairly basic and of folklore character to much extent. Henceforth, we shall not attribute them to any particular source! 1. Preliminaries Let S be a set, idS = {(a, a)| a ∈ S} and irS = (S × S)− idS . Let α be a binary relation defined on S (i.e., α ⊆ S × S). We put i(α) = α ∩ irS and r(α) = α ∪ idS . The relation α is called – irreflexive if α ⊆ irS (equivalently, α ∩ idS = ∅ or i(α) = α); – reflexive if idS ⊆ α (or r(α) = α); – strictly (or sharply) antisymmetric if (a, b) ∈ α implies (b, a) / ∈ α; – antisymmetric if a = b whenever (a, b) ∈ α and (b, a) ∈ α; – symmetric if (a, b) ∈ α implies (b, a) ∈ α; – transitive if (a, c) ∈ α whenever (a, b) ∈ α and (b, c) ∈ α; – a quasiordering if α is reflexive and transitive; – a strict (or sharp) ordering if α is irreflexive and transitive; – a near-ordering if α is antisymmetic and transitive; – a (reflexive) ordering if α is reflexive, antisymmetric and transitive; – a tolerance if α is reflexive and symmetric; – an equivalence if α is reflexive, symmetric and transitive. 1.1. Lemma. Let α be a binary relation on a set S. (i) α is both irreflexive and reflexive iff α = ∅ = S. (ii) α is strictly antisymmetric iff α is irreflexive and antisymmetric. (iii) α is both strictly antisymmetric and symmetric iff α ⊆ idS. (iv) If α is transitive then α is irreflexive iff α is strictly antisymmetric. (v) If α is irreflexive, symmetric and transitive then α = ∅. 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.