Transformation Monoid Research Articles

Hereditarily finitely based monoids of extensive transformations

It is shown that the monoid En of extensive transformations of a chain of order n is hereditarily finitely based if and only if n ≤ 3. It follows that the submonoid OEn of order-preserving transformations in En is also hereditarily finitely based if and only if n ≤ 3.

All countable monoids embed into the monoid of the infinite random graph

We prove that the full transformation monoid on a countably infinite set is isomorphic to a submonoid of End ( R ) , the endomorphism monoid of the infinite random graph R . Consequently, End ( R ) embeds each countable monoid, satisfies no nontrivial monoid identity, and has an undecidable universal theory.

Embedding properties of endomorphism semigroups

Denote by $\mathop{\rm PSelf}\nolimits\varOmega$ (resp., $\mathop{\rm Self}\nolimits\varOmega$) the partial (resp., full) transformation monoid over a set $\varOmega$, and by $\mathop{\rm Sub}\nolimits V$ (resp., $\mathop{\rm End}\nolimits V$) the co

Congruences on monoids of transformations preserving the orientation on a finite chain

The main subject of this paper is the description of the congruences on certain monoids of transformations on a finite chain X n with n elements. Namely, we consider the monoids OR n and POR n of all full, respectively partial, transformations on X n that preserve or reverse the orientation, as well as their respective submonoids OP n and POP n of all orientation-preserving elements. The inverse monoid PORI n of all injective elements of POR n is also considered.

THE VARIETY GENERATED BY A CERTAIN TRANSFORMATION MONOID

It is shown that the transformation monoid [Formula: see text] is finitely based and a finite identity basis for [Formula: see text] is given, which solves the finite basis problem for [Formula: see text]. It is also shown that the variety Var[Formula: see text] generated by [Formula: see text] for which xyx is not an isoterm has uncountably many subvarieties, which gives an affirmative answer to a question of Jackson.

Collapsing monoids consisting of permutations and constants

In this paper we determine all collapsing transformation monoids that contain at least one unary constant operation and whose nonconstant operations are permutations. Furthermore, we find an infinite family of transformation monoids that consist of at least three unary constant operations and some permutations for which the corresponding monoidal intervals are 2-element chains.

The idempotent-separating degree of a block-group

In this note we characterize the least positive integer n such that there exists an idempotent-separating homomorphism from a finite block-group S into the monoid of all partial transformations of a set with n elements. In particular, as for a fundamental semigroup S this number coincides with the smallest size of a set for which S can be faithfully represented by partial transformations, we obtain a generalization of Easdown’s result established for fundamental finite inverse semigroups.

HALL'S CONDITION AND IDEMPOTENT RANK OF IDEALS OF ENDOMORPHISM MONOIDS

AbstractIn 1990, Howie and McFadden showed that every proper two-sided ideal of the full transformation monoid $T_n$, the set of all maps from an $n$-set to itself under composition, has a generating set, consisting of idempotents, that is no larger than any other generating set. This fact is a direct consequence of the same property holding in an associated finite $0$-simple semigroup. We show a correspondence between finite $0$-simple semigroups that have this property and bipartite graphs that satisfy a condition that is similar to, but slightly stronger than, Hall's condition. The results are applied in order to recover the above result for the full transformation monoid and to prove the analogous result for the proper two-sided ideals of the monoid of endomorphisms of a finite vector space.

On groups with every normal subgroup transitive or semiregular

We investigate finite solvable permutation groups in which all normal subgroups are transitive or semiregular. The motivation comes from universal algebra: such groups are examples of collapsing transformation monoids.

Largest subsemigroups of the full transformation monoid

In this paper we are concerned with the following question: for a semigroup S, what is the largest size of a subsemigroup T ⩽ S where T has a given property? The semigroups S that we consider are the full transformation semigroups; all mappings from a finite set to itself under composition of mappings. The subsemigroups T that we consider are of one of the following types: left zero, right zero, completely simple, or inverse. Furthermore, we find the largest size of such subsemigroups U where the least rank of an element in U is specified. Numerous examples are given.

The Minimal Clones above the Permutations

We determine the atoms of the interval of the clone lattice consisting of those clones which contain all permutations, on an infinite base set. This is equivalent to the description of the atoms of the lattice of transformation monoids above the permutations.

Generalized Affine Transformation Monoid of a Residue Class Ring

In this paper, we consider the generalized affine transformation monoid Gaff (A/I) of a residue class ring A/I of a commutative ring A with identity modulo its nonzero ideal I. For the general case, we investigate the Green's relations, Schutzenberger group for every D-class and, the structure of group H-classes, regular D-classes, the idempotent set E(Gaff (A/I)) and the regular element set Reg (Gaff (A/I)) of Gaff (A/I). If A is an integral domain and I a product of powers of invertible maximal ideals, we show that Gaff (A/I) is an epigroup, every R⋆-classes of Gaff (A/I) is a nil-extension of a right group and that Gaff (A/I) is a complete lattice of nil-extensions of right groups.

Collapsing inverse monoids

In this paper we investigate a class of inverse transformation monoids constructed from finite lattices, and we describe a necessary and sufficient condition for such a transformation monoid to be collapsing.

The endomorphism monoid of the random poset contains all countable semigroups

The transformation monoid on a countable set (and hence all countable semigroups) embeds in the endomorphism monoid of the countable homogeneous universal poset, the Fraisse limit of the class of all finite posets.

On the 3-state Mealy automata over an m-symbol alphabet of growth order [formula omitted

We consider sequence { J m , m ⩾ 2 } of 3-state Mealy automata over an m-symbol alphabet such that the growth of J m is intermediate of order [ n log n / 2 log m ] . For each automaton J m we describe the transformation monoid S J m , defined by it, provide generating series for the growth functions, and consider some properties of S J m and J m .

A Presentation of the Singular Part of the Symmetric Inverse Monoid#

We give a semigroup presentation of the singular part of the symmetric inverse monoid on a finite set. Along the way, we derive a monoid presentation of the monoid of all order-preserving injective partial transformations on a finite chain, which differs from the presentation discovered by Fernandes.

Relative Abelian kernels of some classes of transformation monoids

We consider the symmetric inverse monoid ℐn of an n-element chain and its inverse submonoids ℐn, ℐn, ℐn and ℘ℐn of all order-preserving, order-preserving or order-reversing, orientation-preserving and orientation-preserving or orientation-reversing transformations, respectively, and give descriptions of their Abelian kernels relative to decidable pseudovarieties of Abelian groups.

RANK PROPERTIES OF ENDOMORPHISMS OF INFINITE PARTIALLY ORDERED SETS

RANK PROPERTIES OF ENDOMORPHISMS OF INFINITE PARTIALLY ORDERED SETS

Precomplete Clones on Infinite Sets which are Closed under Conjugation

We show that on an infinite set, there exist no other precomplete clones closed under conjugation except those which contain all permutations. Since on base sets of some infinite cardinalities, in particular on countably infinite ones, the precomplete clones containing the permutations have been determined, this yields a complete list of the precomplete conjugation-closed clones in those cases. In addition, we show that there exist no precomplete submonoids of the full transformation monoid which are closed under conjugation except those which contain the permutations; the monoids of the latter kind are known.

The smallest Mealy automaton of intermediate growth

In this paper we study the automaton I 2 , the smallest Mealy automaton of intermediate growth, first considered by the last two authors [I.I. Reznykov, V.I. Sushchansky, The two-state Mealy automata over the two-symbol alphabet of the intermediate growth, Mat. Zametki 72 (2002) 102–117]. We describe the automatic transformation monoid defined by I 2 , give a formula for the generating series for the (ball volume) growth function of I 2 , and give sharp asymptotics for the growth function of I 2 , namely γ I 2 ( n ) ∼ 2 5 / 2 3 3 / 4 π −2 n 1 / 4 exp ( π n / 6 ) , with the ratios of left- to right-hand side tending to 1 as n → ∞ .