We consider various classes of algebras obtained by expanding idempotent semirings with meet, residuals and Kleene-∗. An investigation of congruence properties (epermutability, e-regularity, congruence distributivity) is followed by a section on algebraic Gentzen systems for proving inequalities in idempotent semirings, in residuated lattices, and in (residuated) Kleene lattices (with cut). Fin...

This paper is a brief introduction to idempotent and tropical mathematics. Tropical mathematics can be treated as a result of the so-called Maslov dequantization of the traditional mathematics over numerical fields as the Planck constant ~ tends to zero taking imaginary values. This paper is a brief introduction, practically without exact theorems and proofs, to the Maslov dequantization and id...

Previous work has shown a relation between L-valued extensions of FCA and the spectra of some matrices related to L-valued contexts. We investigate the spectra of reducible matrices over completed idempotent semifields in the framework of dioids, naturally-ordered semirings, that encompass several of those extensions. Considering special sets of eigenvectors also brings out complete lattices in...

Multidimensional minimax single facility location problems with Chebyshev distance are examined within the framework of idempotent algebra. A new algebraic solution based on an extremal property of the eigenvalues of irreducible matrices is given. The solution reduces both unconstrained and constrained location problems to evaluation of the eigenvalue and eigenvectors of an appropriate matrix. ...

We construct a counterexample to Solel’s[25] conjecture that the range of any contractive, idempotent, MASA bimodule map on B(H) is necessarily a ternary subalgebra. Our construction reduces this problem to an analogous problem about the ranges of idempotent maps that are equivariant with respect to a group action. Such maps are important to understand Hamana’s theory of G-injective operator sp...

Modes are idempotent and entropic algebras. More precisely, an algebra (A,Ω) of type τ : Ω −→ Z is called a mode if it is idempotent and entropic, i.e. each singleton in A is a subalgebra and each operation ω ∈ Ω is actually a homomorphism from an appropriate power of the algebra. Both properties can also be expressed by the following identities: (I) ∀ω ∈ Ω, x . . . xω = x (E) ∀ω, φ ∈ Ω, with m...

Idempotent nonlinear filters may be viewed as extensions of the class of (non-realizable) ideal linear filters, and one of their characteristic features is that they reduce any input sequence {x k } to a root after one pass. This paper explores some new constructions of idempotent nonlinear filters, using ideas from algebraic category theory, which provides a natural basis for considering casca...

The problem on the solutions of homogeneous and nonhomogeneous generalized linear vector equations in idempotent algebra is considered. For the study of equations, an idempotent analog of matrix determinant is introduced and its properties are investigated. In the case of irreducible matrix, existence conditions are found and the general solutions of equations are obtained. The results are exte...

Monotone frameworks is one of the most successful frameworks for intraprocedural data flow analysis extending the traditional class of bitvector frameworks (like live variables and available expressions). Weighted pushdown systems is similarly one of the most general frameworks for interprocedural analysis of programs. However, it makes use of idempotent semirings to represent the sets of prope...

We obtain the following characterization of Q-polynomial distance-regular graphs. Let Γ denote a distance-regular graph with diameter d ≥ 3. Let E denote a minimal idempotent of Γ which is not the trivial idempotent E0. Let {θ∗ i }i=0 denote the dual eigenvalue sequence for E. We show that E is Q-polynomial if and only if (i) the entry-wise product E ◦ E is a linear combination of E0, E, and at...