Maximal Subsemigroups Research Articles

Maximal Regular Subsemibands of Finite Orientation-Preserving Partial Transformation Semigroup

Let \(\mathcal {POP}_n\) be the semigroup of all orientation-preserving partial transformations on the finite chain \(X_n=\{1<2<\cdots <n\}\). For \(1\le r\le n-1\), set \(\mathcal {POP}(n, r)=\{\alpha \in \mathcal {POP}_n {:} \, |\mathop {\mathrm {im}}\nolimits (\alpha )|\le r\}\). In this paper, we investigate the maximal regular subsemigroups and the maximal regular subsemibands of the semigroup \(\mathcal {POP}(n,r)\). First, we completely describe the maximal regular subsemigroups of the semigroup \(\mathcal {POP}(n,r)\), for \(2\le r\le n-1\). Secondly, we show that, for \(2\le r \le n-2\), any maximal regular subsemigroup of the semigroup \(\mathcal {POP}(n,r)\) is a semiband and prove that the maximal regular subsemigroups and the maximal regular subsemibands of the semigroup \(\mathcal {POP}(n,r)\) coincide, for \(2\le r\le n-2\). Finally, we obtain the complete classification of maximal regular subsemibands of the semigroup \(\mathcal {POP}(n,n-1)\).

Maximal Regular Subsemibands of the Finite Order-Preserving Partial Transformation Semigroups $$\mathcal {PO}(n,r)$$ PO ( n , r )

Let $$\mathcal {PO}_n$$ be the semigroup of all order-preserving partial transformations on the finite set $$X_n=\{1, 2,\ldots , n\}$$ . For $$1\le r\le n-1$$ , set $$\mathcal {PO}(n, r)=\{\alpha \in \mathcal {PO}_n: |\mathop {\text{ im }}\nolimits (\alpha )|\le r\}$$ . In this paper, we investigate the maximal regular subsemigroups and the maximal regular subsemibands of the semigroup $$\mathcal {PO}(n,r)$$ . First, we completely describe the maximal regular subsemigroups of the semigroup $$\mathcal {PO}(n,r)$$ , for $$1\le r\le n-1$$ . Secondly, we show that, for $$2\le r \le n-2$$ , any maximal regular subsemigroup of the semigroup $$\mathcal {PO}(n,r)$$ is a semiband and obtain that the maximal regular subsemigroups and the maximal regular subsemibands of the semigroup $$\mathcal {PO}(n,r)$$ coincide, for $$2\le r\le n-2$$ . Finally, we obtain the complete classification of maximal regular subsemibands of the semigroup $$\mathcal {PO}_n$$ .

Maximal inverse subsemigroups of the semigroup of all full transformations satisfying x ≈ x3

We classify the maximal Clifford inverse subsemigroups [Formula: see text] of the full transformation semigroup [Formula: see text] on an [Formula: see text]-element set with [Formula: see text] for all [Formula: see text]. This classification differs from the already known classifications of Clifford inverse semigroups, it provides an algorithm for its construction. For a given natural number [Formula: see text], we find also the largest size of an inverse subsemigroup [Formula: see text] of [Formula: see text] satisfying [Formula: see text] with least rank [Formula: see text] for any element in [Formula: see text].

Maximal subsemigroups containing a particular semigroup

Abstract We characterize maximal subsemigroups of the monoid T(X) of all transformations on the set X = ℕ of natural numbers containing a given subsemigroup W of T(X) such that T(X) is finitely generated over W. This paper gives a contribution to the characterization of maximal subsemigroups on the monoid of all transformations on an infinite set.

Maximal subsemigroups of the semigroup of all mappings on an infinite set

In this paper we classify the maximal subsemigroups of the full transformation semigroup Ω Ω \Omega ^\Omega , which consists of all mappings on the infinite set Ω \Omega , containing certain subgroups of the symmetric group Sym ⁡ ( Ω ) \operatorname {Sym}(\Omega ) on Ω \Omega . In 1965 Gavrilov showed that there are five maximal subsemigroups of Ω Ω \Omega ^\Omega containing Sym ⁡ ( Ω ) \operatorname {Sym}(\Omega ) when Ω \Omega is countable, and in 2005 Pinsker extended Gavrilov’s result to sets of arbitrary cardinality. We classify the maximal subsemigroups of Ω Ω \Omega ^\Omega on a set Ω \Omega of arbitrary infinite cardinality containing one of the following subgroups of Sym ⁡ ( Ω ) \operatorname {Sym}(\Omega ) : the pointwise stabiliser of a non-empty finite subset of Ω \Omega , the stabiliser of an ultrafilter on Ω \Omega , or the stabiliser of a partition of Ω \Omega into finitely many subsets of equal cardinality. If G G is any of these subgroups, then we deduce a characterisation of the mappings f , g ∈ Ω Ω f,g\in \Omega ^\Omega such that the semigroup generated by G ∪ { f , g } G\cup \{f,g\} equals Ω Ω \Omega ^\Omega .

A finite interval in the subsemigroup lattice of the full transformation monoid

In this paper we describe a portion of the subsemigroup lattice of the full transformation semigroup Ω Ω , which consists of all mappings on the countable infinite set Ω. Gavrilov showed that there are five maximal subsemigroups of Ω Ω containing the symmetric group $\operatorname {Sym}(\varOmega )$ . The portion of the subsemigroup lattice of Ω Ω which we describe is that between the intersection of these five maximal subsemigroups and Ω Ω . We prove that there are only 38 subsemigroups in this interval, in contrast to the $2^{2^{\aleph_{0}}}$ subsemigroups between $\operatorname {Sym}(\varOmega )$ and Ω Ω .

A note on maximal regular subsemigroups of the finite transformation semigroups $\mathcal{T}(n,r)$

Let \(\mathcal{T}_{n}\) be the semigroup of all full transformations on the finite set Xn={1,2,…,n}. For 1≤r≤n, set \(\mathcal {T}(n, r)=\{ \alpha\in\mathcal{T}_{n} | \operatorname{rank}(\alpha)\leq r\}\). In this note we show that, for 2≤r≤n−2, any maximal regular subsemigroup of the semigroup \(\mathcal{T} (n,r)\) is idempotent generated, but this may not happen in the semigroup \(\mathcal{T}(n, n-1)\).

Locally Maximal Idempotent-generated Subsemigroups of Finite Orientation-preserving Singular Partial Transformation Semigroups

In this paper we describe locally maximal idempotent-generated subsemigroups of finite orientation-preserving singular partial transformation semigroups SPOPn and obtain a complete classification. We also obtain a classification of locally maximal idempotent-generated subsemigroups of finite order-preserving singular partial transformation semigroups [Formula: see text] with respect to ≤k.

SEMIGROUP OF INJECTIVE PARTIAL TRANSFORMATIONS WITH BOUNDED GAP AND FIXED DEFECT

Let X be an infinite set and suppose that ℵ0 ≤ q ≤ |X|. In 2004, Pinto and Sullivan considered algebraic properties of PS(q), the partial Baer-Levi semigroup consisting of all injective partial transformations α of X such that |X \Xα| = q. They also determined its subsemigroup S(q,r) = {α ∈ PS(q) : |X domα| ≤ r} where ℵ0 ≤ r ≤ |X|. Recently, Singha and Sanwong showed that, when q < |X|, almost every maximal subsemigroup of PS(q) is induced by a maximal subsemigroup of S(q,r). Here, we use their work to describe some algebraic properties of S(q,r) including its Green's relation and its ideal structure.

The maximal subsemigroups of semigroups of transformations preserving or reversing the orientation on a finite chain

The study of the semigroups OPn, of all orientation-preserving transformations on an n-element chain, and ORn, of all orientation-preserving or orientation-reversing transformations on an n-element chain, has began in [17] and [5]. In order to bring more insight into the subsemigroup structure of OPn and ORn, we characterize their maximal subsemigroups.

On the Monoid of All Partial Order-Preserving Extensive Transformations

A partial transformation α on an n-element chain X n is called order-preserving if x ≤ y implies xα ≤yα for all x, y in the domain of α and it is called extensive if x ≤ xα for all x in the domain of α. The set of all partial order-preserving extensive transformations on X n forms a semiband POE n . We determine the maximal subsemigroups as well as the maximal subsemibands of POE n .

A Note on Maximal Properties of Some Subsemigroups of Finite Order-Preserving Transformation Semigroups

Both maximal idempotent-generated subsemigroups and maximal idempotent-generated regular subsemigroups of O n were studied by Yang [10]. The purpose of this article is to simplify the results of Yang [10].

A classification of maximal idempotent-generated subsemigroups of finite orientation-preserving singular partial transformation semigroups

In this paper we describe the maximal idempotent-generated subsemigroups of the finite orientation-preserving singular partial transformation semigroup SPOPn and obtain their complete classification. We also obtain a classification of the maximal idempotent-generated subsemigroups of the finite order-preserving singular partial transformation semigroup \(\mathit{PO}^{k}_{n}\) with respect to ≤k.

On nilpotent subsemigroups of the matrix semigroup over an antiring

In this paper, nilpotent subsemigroups in the matrix semigroup over a commutative antiring are discussed. Some basic properties and characterizations for the nilpotent subsemigroups are given, and some equivalent conditions for the matrix semigroup over a commutative antiring to have a maximal nilpotent subsemigroup are obtained. Also, the maximal nilpotent subsemigroups in the matrix semigroup are described.

On Maximal Subsemigroups of Partial Baer‐Levi Semigroups

Suppose that X is an infinite set with |X | ≥ q ≥ ℵ0 and I(X) is the symmetric inverse semigroup defined on X. In 1984, Levi and Wood determined a class of maximal subsemigroups MA (using certain subsets A of X) of the Baer‐Levi semigroup BL(q) = {α ∈ I(X): dom α = X and |X∖Xα| = q}. Later, in 1995, Hotzel showed that there are many other classes of maximal subsemigroups of BL(q), but these are far more complicated to describe. It is known that BL(q) is a subsemigroup of the partial Baer‐Levi semigroup PS(q) = {α ∈ I(X) : |X∖Xα| = q}. In this paper, we characterize all maximal subsemigroups of PS(q) when |X | &gt; q, and we extend MA to obtain maximal subsemigroups of PS(q) when |X | = q.

On the maximal regular subsemigroups of ideals of order-preserving or order-reversing transformations

We characterize the maximal regular subsemigroups of the ideals of the semigroup of all order-preserving transformations as well as of the semigroup of all order-preserving or order-reversing transformations on a finite ordered set.

The Ideal Structure of Semigroups of Linear Transformations with Lower Bounds on Their Nullity or Defect

Suppose V is an infinite-dimensional vector space and let T(V) denote the semigroup (under composition) of all linear transformations of V. In this paper, we study the semigroup OM(p,q) consisting of all α ∈ T(V) for which dim ker α ≥ q and the semigroup OE(p,q) of all α ∈ T(V) for which codim ran α ≥ q, where dim V = p ≥ q ≥ ℵ0. It is not difficult to see that OM(p,q) and OE(p,q) are a right ideal and a left ideal of T(V), respectively, and using these facts, we show that they belong to the class of all semigroups whose sets of bi-ideals and quasi-ideals coincide. Also, we describe Green's relations and the two-sided ideals of each semigroup, and determine its maximal regular subsemigroup. Finally, we determine some maximal right cancellative subsemigroups of OE(p,q).

Structure of the semigroup OT n

We study the structure of the semigroup OT n , which is a unique (up to an isomorphism) R-section of the semigroup T n . For this semigroup, we describe Green relations, determine regular and nilpotent elements, describe maximal nilpotent subsemigroups, and determine the unique irreducible system of generatrices and maximal subsemigroups.

Mathematical modeling of nilpotent subsemigroups of semigroups of contracting transformations of a Boolean

We study mathematical models of the structure of nilpotent subsemigroups of the semigroup PTD(B n ) of partial contracting transformations of a Boolean, the semigroup TD(B n ) of full contracting transformations of a Boolean, and the inverse semigroup ISD(B n ) of contracting transformations of a Boolean. We propose a convenient graphical representation of the semigroups considered. For each of these semigroups, the uniqueness of its maximal nilpotent subsemigroup is proved. For PTD(B n ) and TD(B n ) , the capacity of a maximal nilpotent subsemigroup is calculated. For ISD(B n ), we construct estimates for the capacity of a maximal nilpotent subsemigroup and calculate this capacity for small n. For all indicated semigroups, we describe the structure of nilelements and maximal nilpotent subsemigroups of nilpotency degree k and determine the number of elements and subsemigroups for some special cases.

A classification of maximal idempotent-generated subsemigroups of singular orientation-preserving transformation semigroups

We describe maximal idempotent-generated subsemigroups of finite singular orientation-preserving transformation semigroups and completely obtain their classification.