A language L ⊆ A∗ is literally idempotent in case that uav ∈ L if and only if uav ∈ L, for each u, v ∈ A∗, a ∈ A. We already studied classes of such languages closely related to the (positive) varieties of the famous StraubingThérien hierarchy. In the present paper we start a systematic study of literal varieties of literally idempotent languages, namely we deal with the case of two letter alph...

We show that every (finite or not) typing derivation of system M, using non-idempotent intersection, which is the infinitary version of de Carvalho’s system M0, can be represented in a rigid, non-idempotent intersection type system S. Namely, whereas non-idempotent intersection is represented by multisets in system M, system S resort to families of types indexed by integers, called tracks. The ...

We construct the freest idempotent-pure expansion of an inverse semigroup, generalizing an expansion of Margolis and Meakin for the group case. We also generalize the Birget-Rhodes prefix expansion to inverse semigroups with an application to partial actions of inverse semigroups. In the process of generalizing the latter expansion, we are led to a new class of idempotent-pure homomorphisms whi...

Cyclic codes are very useful in error correcting codes. Hence, the construction of good cyclic codes is of utmost importance. A generic class of polynomials called generator polynomials is used to construct the cyclic codes. Finding a generator polynomial necessitates factoring a base polynomial called monic polynomial. Unfortunately, factoring a polynomial is not always an easy task. Idempoten...

A graph G is said to be hom-idempotent if there is a homomorphism from G2 to G, and weakly hom-idempotent if for some n ≥ 1 there is a homomorphism from Gn+1 to Gn . We characterize both classes of graphs in terms of a special class of Cayley graphs called normal Cayley graphs. This allows us to construct, for any integer n, a Cayley graph G such that Gn+1 → Gn 6→ Gn−1, answering a question of ...

A game is considered where the communication network of the first player is explicitly modeled. The second player may induce delays in this network, while the first player may counteract such actions. Costs are modeled through expectations over idempotent probability measures. The idempotent probabilities are conditioned by observational data, the arrival of which may have been delayed along th...

In this paper we investigate the atomic level in the lattice of subvarieties of residuated lattices. In particular, we give infinitely many commutative atoms and construct continuum many non-commutative, representable atoms that satisfy the idempotent law; this answers Problem 8.6 of [12]. Moreover, we show that there are only two commutative idempotent atoms and only two cancellative atoms. Fi...

Idempotent integration is an analogue of the Lebesgue integration where σ-additive measures are replaced by σ-maxitive measures. It has proved useful in many areas of mathematics such as fuzzy set theory, optimization, idempotent analysis, large deviation theory, or extreme value theory. Existence of Radon–Nikodym derivatives, which turns out to be crucial in all of these applications, was prov...

We consider subsemimodules and convex subsets of semimodules over semirings with an idempotent addition. We introduce a nonlinear projection on subsemimodules: the projection of a point is the maximal approximation from below of the point in the subsemimodule. We use this projection to separate a point from a convex set. We also show that the projection minimizes the analogue of Hilbert’s proje...