It is well known that intra-regular semigroups (ordered semigroups) play an essential role in studying the structure of semigroups (ordered semigroups). In this paper we present a structure theorem referring to the decomposition of ordered semigroups into left strongly simple components, that is, into components, which are both simple and left quasi-regular. We prove, among others, that if an o...

Given a commutative semigroup S with 0, where 0 is the unique singleton ideal, we associate a simple graph Γ(S), whose vertices are labeled with the nonzero elements in S. Two vertices in Γ(S) are adjacent if and only if the corresponding elements multiply to 0. The inverse problem, i.e., given an arbitrary simple graph, whether or not it can be associated to some commutative semigroup, has pro...

The classical approach of solvability using group theory is well known and one original motivation is to solve polynomials by radicals. Radicals are square, cube, square root, cube root etc of the original coefficients for the polynomial. A polynomial is solvable by radicals if the permutation group is solvable. This is exact solvability via group theory. With modern computers, we might need to...

A generalization of the original Diffie-Hellman key exchange in (Z/pZ) found a new depth when Miller [27] and Koblitz [16] suggested that such a protocol could be used with the group over an elliptic curve. In this paper, we propose a further vast generalization where abelian semigroups act on finite sets. We define a Diffie-Hellman key exchange in this setting and we illustrate how to build in...

Constant Rank (CR) state machines play an important role in the general structure theory of Finite State Machines. A machine is of constant rank if each input and input-sequence maps the state set onto the same number of next states. CR-machines are analysed via their sequential closure (semigroup), which is a simple semigroup : a semidirect product (L×R) ∗G of a leftand a right-copy semigroup,...

The main purpose of this paper is to show that a regular semigroup S is a semilattice of bisimple semigroups if and only if it is a band of bisimple semigroups and that this holds if and only if 3) is a congruence on S. It is also shown that a quasiregular semigroup 5 which is a rectangular band of bisimple semigroups is itself bisimple. In [3, Theorem 4.4] it was shown that a semigroup S is a ...

Goes to the advanced topics rather fast. Contains up-to-date proofs for free inverse semigroups, topics on biordered sets, Isbell's zigzags , and some combinatorics of transformation semigroups.] Fundamentals of semigroup theory, 1995. [Probably the best introduction. Contains more advanced material such as Isbell's zigzags , and the free inverse semi-groups]. G. Lallement, Semigroups and combi...

A (v, b, r, k) combinatorial configuration is a (r, k)-biregular bipartite graph with v vertices on the left and b vertices on the right and with no cycle of length 4. Combinatorial configurations have become very important for some cryptographic applications to sensor networks and to peerto-peer communities. Configurable tuples are those tuples (v, b, r, k) for which a (v, b, r, k) combinatori...