The numbers of distinct self-orthogonal Latin squares (SOLS) and idempotent SOLS have been enumerated for orders up to and including 9. The isomorphism classes of idempotent SOLS have also been enumerated for these orders. However, the enumeration of the isomorphism classes of non-idempotent SOLS is still an open problem. By utilising the automorphism groups of class representatives from the al...

The idempotent problem of a finitely generated inverse semi-group is the formal language of all words over the generators representing idempotent elements. This note proves that a finitely generated inverse semigroup with regular idempotent problem is necessarily finite. This answers a question of Gilbert and Noonan Heale, and establishes a gen-eralisation to inverse semigroups of Anisimov's Th...

The author’s modification of Eilenberg theorem relates the so-called conjunctive varieties of regular languages with pseudovarieties of idempotent semirings. Recent results by Pastijn and his co-authors lead to the description of the lattice of all (pseudo)varieties of idempotent semirings with idempotent multiplication. We describe here the corresponding 78 varieties of languages.

An approach to schedule development in project management is proposed based on models and methods of idempotent algebra. The approach offers a way to represent various types of precedence relationships among activities in projects as linear vector equations in terms of an idempotent semiring. As a result, many issues in project scheduling reduce to solving computational problems in the idempote...

Semiheaps are ternary generalisations of involuted semigroups. The first kind of semiheaps studied were heaps, which correspond closely to groups. We apply the radical theory of varieties of idempotent algebras to varieties of idempotent semiheaps. The class of heaps is shown to be a radical class, as are two larger classes having no involuted semigroup counterparts. Radical decompositions of v...

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.