Presenting Finite Posets

String rewriting systems have been originally introduced by Thue [21] in order to study word problems in monoids. A string rewriting system (Σ,R) consists of a set Σ, called the alphabet, and a set R ⊆ Σ∗×Σ∗ of rules. The monoid Σ∗/≡R, obtained by quotienting the free monoid Σ∗ over Σ by the smallest congruence (wrt concatenation) containing R, is called the monoid presented by the rewriting system. The rewriting system can thus be thought of as a small description of the monoid, and the word problem consists in deciding whenever two words u,v ∈ Σ∗ represent the same word, i.e. are such that u≡R v. Now, when the rewriting system is convergent, i.e. both terminating and confluent, normal forms provide canonical representatives of the equivalence classes: two words u,v ∈ Σ∗ are equivalent by the congruence ≡R if and only if they have the same normal form, and the word problem can be thus be decided in this case. Example 1. Consider the rewriting system (Σ,R) with Σ = {a,b} and R = {(ba,ab),(bb,ε)}, where ε denotes the empty word. This rewriting system is easily shown to be terminating, and the two critical pairs can be joined: bba {{ !! bab // abb // a bbb