The concept of idempotent elements plays an important role in semigroup theory and semiring theory. In this paper we characterize idempotent pre-generalized hypersubstitutions of type τ = (2, 2).

A unifying approach to software and hardware design generated by ideas of Idempotent Mathematics is discussed. The so-called idempotent correspondence principle for algorithms, programs and hardware units is described. A software project based on this approach is presented.

In this paper, we present necessary and sufficient conditions for X? to be idempotent orthogonal idempotent, where ? {A?, AD, AD?, A?,D Aw}. Several characteristics when is are derived by core-EP decomposition. Additionally, give some equivalent matrix A using the properties of generalized inverses A.

In a previous paper we introduce the concept of full d-stability, in this work several types of generalizations were introduced ; minimal (maximal) d-stable; fully pseudo d-stable and afd-stable module. A dual to the notion of terse module is, also, introduced namely d-terse and it is shown that it is coincide with fully pseudo d-stable.

let $r$ be a right artinian ring or a perfect commutative‎‎ring‎. ‎let $m$ be a noncosingular self-generator $sum$-lifting‎‎module‎. ‎then $m$ has a direct decomposition $m=oplus_{iin i} m_i$‎,‎where each $m_i$ is noetherian quasi-projective and each‎‎endomorphism ring $end(m_i)$ is local‎.