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...

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 give a necessary and sufficient condition for the preservation of finite products by a reflection of a variety of universal algebras into an idempotent subvariety. It is also shown that simple and semi-left-exact reflections into subvarieties of universal algebras are the same. It then follows that a reflection of a variety of universal algebras into an idempotent subvariety has stable units...