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A V-labelled poset P can induce an operation on the languages on any fixed alphabet, as well as an operation on labelled posets (as noticed by Pratt and Gischer (Pratt 1986; Gischer 1988)). For any collection X of V-labelled posets and any alphabet Σ we obtain an X-algebra ΣX of languages on Σ. We consider the variety Lang(X) generated by these algebras when X is a collection of nonempty ‘traceable posets’. The current paper contains several observations about this variety. First, we use one of the basic results in Bloom and Ésik (1996) to show that a concrete description of the A-generated free algebra in Lang(X) is the X-subalgebra generated by the singletons (labelled a∈A) in the X-algebra of all A-labelled posets. Equipped with an appropriate ordering, these same algebras are the free ordered algebras in the variety Lang(X)[les ] of ordered language X-algebras. Further, if one enriches the language algebras by adding either a binary or infinitary union operation, the free algebras in the resulting variety are described by certain ‘closed’ subsets of the original free algebras. Second, we show that for ‘reasonable sets’ X, the variety Lang(X) has the property that for each n[ges ]2, the n-generated free algebra is a subalgebra of the 1-generated free algebra. Third, knowing the free algebras enables us to show that these varieties are generated by the finite languages on a two-letter alphabet.

We show that, aside from the semiring equations, three equations and two equation schemes characterize the semiring of regular sets with the Kleene star operation.