Given a monad T on Set whose functor factors through the category of ordered sets with left adjoint maps, the category of Kleisli monoids is defined as the category of monoids in the hom-sets of the Kleisli category of T. The Eilenberg-Moore category of T is shown to be strictly monadic over the category of Kleisli monoids. If the Kleisli category of T moreover forms an order-enriched category,...

We study generalized inverses on semigroups by means of Green’s relations. We first define the notion of inverse along an element and study its properties. Then we show that the classical generalized inverses (group inverse, Drazin inverse and Moore-Penrose inverse) belong to this class. There exist many specific generalized inverses in the literature, such as the group inverse, the Moore-Penro...

Mathematical Morphology arose in 1964 by the work of George Matheron and Jean Serra, who developed its main concepts and tools. It uses concepts from algebra and geometry. (Set theory, complete lattice theory, convexity etc,). The notion of adjunction is very fundamental in Mathematical Morphology. Morphological systems is a broad class of nonlinear signal operators that have found many applica...

A dynamical systems based model of computation is studied. We demonstrate the computational capability of a class of dynamical systems called switching map systems. There exists a switching map system with two types of baker's map to emulate any Turing machines. The baker's maps are corresponding to the elementary operations of Turing machines such as left/right head-moving and read/write symbo...

We define the class of sequentially semi-separable matrices in this paper. Essentially this is the class of matrices which have low numerical rank on their off diagonal blocks. Examples include banded matrices, semi-separable matrices, their sums as well as inverses of these sums. Fast and stable algorithms for solving linear systems of equations involving such matrices and computing Moore-Penr...

In this paper we identify conditions under which the cohomology H∗(ΩMξ; k) for the loop space ΩMξ of the Thom space Mξ of a spherical fibration ξ ↓ B can be a polynomial ring. We use the Eilenberg–Moore spectral sequence which has a particularly simple form when the Euler class e(ξ) ∈ H(B; k) vanishes, or equivalently when an orientation class for the Thom space has trivial square. As a consequ...

The following text is an outline of a review paper on Moore and Seiberg’s equations, Topological (Projective) Field Theories in three dimensions and their relationship with Grothendieck’s second paragraph of the “Esquisse d’un Programme”. Because of schedule and length problems this review paper has not been included in this volume. First of all, we recall the construction of projective topolog...

An algorithm for computing {2, 3}, {2, 4}, {1, 2, 3}, {1, 2, 4}-inverses and the Moore-Penrose inverse of a given rational matrix A is established. Classes A{2, 3}s and A{2, 4}s are characterized in terms of matrix products (R∗A)†R∗ and T ∗(AT ∗)†, where R and T are rational matrices with appropriate dimensions and corresponding rank. The proposed algorithm is based on these general representat...