Special idempotents and projections

Zero-Sum Triangles for Involutory, Idempotent, Nilpotent and Unipotent Matrices

In some matrix formations, factorizations and transformations, we need special matrices with some properties and we wish that such matrices should be easily and simply generated and of integers. In this paper, we propose a zero-sum rule for the recurrence relations to construct integer triangles as triangular matrices with involutory, idempotent, nilpotent and unipotent properties, especially nilpotent and unipotent matrices of index 2. With the zero-sum rule we also give the conditions for the special matrices and the generic methods for the generation of those special matrices. The generated…

Determinants and Limit Systems in some Idempotent and Non-Associative Algebraic Structure

This paper considers an idempotent and symmetrical algebraic structure as well as some closely related concept. A special notion of determinant is introduced and a Cramer formula is derived for a class of limit systems derived from the Hadamard matrix product and we give the algebraic form of a sequence of hyperplanes passing through a finite number of points. Thereby, some standard results arising for Max-Times systems with nonnegative entries appear as a special case. The case of two sided systems is also analyzed. In addition, a notion of eigenvalue in limit is considered. It is shown that …

Idempotent Analysis, Tropical Convexity and Reduced Divisors

We investigate a canonical extension of a conventional combinatorial notion of reduced divisors to a notion of tropical projections, which can be defined as the unique minimizers of the so-called $B$-pseudonorms with respect to compact tropical convex sets. In this paper, we build the foundation of a theory of idempotent analysis using tropical projections and obtain a series of subsequent results, e.g. tropical retracts, construction of compact tropical convex sets and a set-theoretical characterization of tropical weak independence. In particular, we prove a tropical version of Mazur's Theor…

Small clones and the projection property

In 1986, the second author classified the minimal clones on a finite universe into five types. We extend this classification to infinite universes and to multiclones. We show that every non-trivial clone contains a "small" clone of one of the five types. From it we deduce, in part, an earlier result, namely that if $\mathcal C$ is a clone on a universe $A$ with at least two elements, that contains all constant operations, then all binary idempotent operations are projections and some $m$-ary idempotent operation is not a projection some $m\geq 3$ if and only if there is a Boolean group $G$ on …

A projection construction for semifields

Simple constructions are given for finite semifields that include as special cases both old semifields and recently constructed semifields.

半域上的表示论

Representation theory over semifields

本文研究挠群在幂等半域（尤其是 Boolean 半域）上的表示并加以分类，为后续推广到低秩拟阵表示奠定基础。