Symmetric Semigroup Research Articles

Automaton-program implementation of a symmetric semigroup. Part I

Automaton-program implementation of a symmetric semigroup. Part I

Distribution of the Number of Fixed Points of Elements of a Symmetric Semigroup under the Condition $\sigma ^{h + 1} = \sigma ^h $ and the Number of Trees of Height Not Exceeding h

Distribution of the Number of Fixed Points of Elements of a Symmetric Semigroup under the Condition $\sigma ^{h + 1} = \sigma ^h $ and the Number of Trees of Height Not Exceeding h

Mutants in the symmetric semigroups

Mutants in the symmetric semigroups

Über den Untergruppenverband der Symmetrischen Gruppe, den Unterhalbgruppenverband der Symmetrischen Halbgruppe und den Unteralgebrenverband der Postschen Algebra

This paper concerns composition chains (i.e. maximal chains) in lattices of subalgebras of function algebras indicated in the title. The description of the shortest composition chains of subsemigroups of the symmetric semigroup Ωκ is reduced to the description of the shortest composition chains of subgroups of the symmetric groupGκ. Boundaries and the structure of spectra of lengths of composition chains are investigated. It is shown that there is no lower boundary of the predicate characterization in the lattice of subalgebras of E. Post's finite-rank algebra.

Some combinatorial properties of transformations and their connections with the theory of graphs

Although many results concerning permutations and permutation groups are known, less attention has been paid to transformations and transformation semigroups. It is true that every abstract group is isomorphic to a permutation group, so that with respect to structure there is not difference between abstract groups and permutation groups. Similarly every abstract semigroup is isomorphic to a transformation semigroup; this has led the author to write some papers on the subject [4, 5]. We restrict ourselves to the finite case, and the aim of this paper is to obtain results in this field by a one-to-one correspondence between transformations and directed graphs. The main results of this paper are as follows: (1) Generalization of the Cauchy formula concerning the number of permutations of degree n with prescribed lengths of cycles. (2) Determination of the number of element triples which are generating systems of the symmetric semigroup of degree n, i.e., the semigroup containing every transformation of degree n. (3) Determination of the expected value of the degree of the main permutation of a random transformation of degree n.

A characterization of idempotents in semigroups

The purpose of this note is to present a theorem which characterizes, exhibits, and counts the number of idempotents in a symmetric groupoid. The proof is simple and it simplifies similar results in two papers by Harris in which the number of idempotents is counted by graph theoretic terminology and by generating functions. Some immediate corollaries give other combinatorial results in semigroup theory as well as identifying subgroups of the symmetric semigroup.

Asymptotic expansions for the coefficients of analytic functions

Abstract : Under suitable hypotheses concerning the analytic function f(z) = Sum from n=o to infinity of (alpha sub n)z to the nth power, it is shown that alpha sub n has an asymptotic expansion for large n. This result is used to estimate the number of idempotent elements in the symmetric semigroup on n elements. (Author)

A Note on the Number of Idempotent Elements in Symmetric Semigroups

Abstract : An alternative derivation of the exponential generating function of the number of idempotent elements in symmetric semigroups is given. (Author)

The number of idempotent elements in symmetric semigroups

This paper counts the number of idempotent elements in the symmetric semigroup on n elements. The exponential generating function is obtained and a number of combinatorial identities and congruences are given. An asymptotic formula is provided along with some numerical comparisons between the asymptotic formula and the exact values.