Finite Codes over Free Binoids

A binoid is an algebra which has two associative operations and the same identity to both operations. For any finite alphabet Σ, Σ*(^, •) denotes the free binoid generated by Σ with two independent associative operations ^ and • and the identity λ. We introduce the notion of two types of finite codes (^-codes and •-codes) over free binoids and show that for any given finite subset X of Σ*(^, •) and x ∈ [^, •], one can decide effectively whether X is a x-code or not.