Narrow Results

In the 1960s, G. Grätzer introduced the notion of the minimal extension property (MEP) of a finite sequence in order to investigate p_n-sequences and free spectra of algebras. While there are many particular results on the MEP, stating that some sequences or families of sequences have the MEP, no general result has been obtained so far and the main general problems remain open.

In this paper we prove a fact about existence of finite complete systems of subalgebras, which generalizes a method used occasionally to prove the minimal extension property of a sequence. Using this we obtain some general results about minimal extensions and, along the way, a rather unexpected solution of one of the central open problems in the area.

The structure of simple and directly indecomposable members of the variety RICRL of representable idempotent commutative residuated lattices is described. These descriptions are employed to determine the size and structure of free algebras in RICRL, using existing theory on the Fraser–Horn and Apple properties. A formula for the free spectrum of RICRL is given.

A semigroup S is said to be ℓ-threshold k-testable if it satisfies all identities u = v where u, v is an arbitrary pair of words over a finite alphabet Σ such that they simultaneously belong or fail to belong to any ℓ-threshold k-testable (regular) language. We give an asymptotic formula for the free spectrum of the variety of all ℓ-threshold k-testable semigroups, thereby providing an asymptotic upper bound on the size of an arbitrary finitely generated locally threshold testable semigroup. The combinatorial interpretation of this task yields an enumeration problem for particular edge labelings of de Bruijn graphs.