A linear vector equation is considered defined in terms of idempotent mathematics. To solve the equation, we apply an approach that is based on the analysis of distances between vectors in idempotent vector spaces and reduces the solution of the equation to that of a tropical optimization problem. Based on the approach, existence and uniqueness conditions are established for the solution, and a...

This paper is a very brief introduction to idempotent mathematics and related topics. It appears as an introductory paper in the volume Idempotent Mathematics and Mathematical Physics (G. L. Litvinov and V. P. Maslov, eds; AMS Contemporary Mathematics Proceedings Series, 2005) [73]. This paper is a very brief introduction, without exact theorems and proofs, to the Maslov dequantization and idem...

We study the singular part of the partition monoid Pn; that is, the ideal Pn \ Sn, where Sn is the symmetric group. Our main results are presentations in terms of generators and relations, and we also show that Pn \Sn is idempotent generated, and that its rank and idempotent-rank are both equal to ( n+1 2 ) = 12n(n + 1). One of our presentations uses an idempotent generating set of this minimal...

In this paper, the notion of fully idempotent modules is defined and it is shown that this notion inherits most of the essential properties of the usual notion of von Neumann's regular rings. Furthermore, we introduce the dual notion of fully idempotent modules (that is, fully coidempotent modules) and investigate some properties of this class of modules.

Tian and Styan have shown many rank equalities for the sum of two and three idempotent matrices and pointed out that rank equalities for the sum P₁ + ⋯+P k with P₁,…, P k be idempotent (k > 3) are still open. In this paper, by using block Gaussian elimination, we obtained rank equalities for the sum of finitely many idempotent matrices and then solved the open problem mentioned above. Extension...

We describe a part of the lattice of subvarieties of left distributive left idempotent groupoids (i.e. those satisfying the identities x(yz) ≈ (xy)(xz) and (xx)y ≈ xy) modulo the lattice of subvarieties of left distributive idempotent groupoids. A free groupoid in a subvariety of LDLI groupoids satisfying an identity x ≈ x decomposes as the direct product of its largest idempotent factor and a ...

The purpose of this paper is to investigate pure submodules of multiplication modules. We introduce the concept of idempotent submodule generalizing idempotent ideal. We show that a submodule of a multiplication module with pure annihilator is pure if and only if it is multiplication and idempotent. Various properties and characterizations of pure submodules of multiplication modules are consid...

We introduce the class of quasi-square-2-isometric operators on a complex separable Hilbert space. This extends 2-isometric due to Agler and Stankus. An operator T is said be if T*5T5 ? 2T*3T3 + T*T = 0. In this paper, we give matrix representation in order obtain spectral properties operator. particular, show that function continuous all operators. Under hypothesis ?(T)?(??(T)) ?, also prove E...

This paper is a very brief introduction to idempotent mathematics and related topics. It appears as an introductory paper in the volume Idempotent Mathematics and Mathematical Physics (G. L. Litvinov and V. P. Maslov, eds; AMS Contemporary Mathematics Proceedings Series, 2005) [73]. This paper is a very brief introduction, without exact theorems and proofs, to the Maslov dequantization and idem...

Path algebras are additively idempotent semirings and generalize Boolean algebras, fuzzy algebras, distributive lattices and inclines. Thus the Boolean matrices, the fuzzy matrices, the lattice matrices and the incline matrices are prototypical examples of matrices over path algebras. In this paper, generalized fuzzy matrices are considered as matrices over path algebras. Compositions of genera...