Sn-Normal Semigroups of Partial Transformations

2012 ◽  
Vol 19 (spec01) ◽  
pp. 947-970 ◽  
Author(s):  
Sean V. Droms ◽  
Janusz Konieczny ◽  
Roberto Palomba
Keyword(s):  
Symmetric Group ◽  
Classification Theorem ◽  
Group Of Units ◽  
Normal Subsemigroup

For an integer n ≥ 1, let [Formula: see text] and Sn be, respectively, the semigroup of partial transformations and the symmetric group on the set X = {1,…,n}. Then Sn is the group of units of [Formula: see text]. A subsemigroup S of [Formula: see text] is Sn-normal if for all a ∈ S and g ∈ Sn, g-1ag ∈ S. In 1976, Symons described the Sn-normal semigroups of full transformations of X. In 1995, Lipscomb and the second author determined the Sn-normal semigroups of partial injective transformations of X. In this paper, we complete the classification by describing all Sn-normal subsemigroups of [Formula: see text]. As a consequence of the classification theorem, we obtain a characterization of the automorphisms of any Sn-normal subsemigroup of [Formula: see text].

scholarly journals Generating infinite monoids of cellular automata

2021 ◽  
pp. 2250215
Author(s):  
Alonso Castillo-Ramirez
Keyword(s):  
Cellular Automata ◽  
Normal Subgroups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Group Of Units ◽  
Finite Dimensional ◽  
Index Condition ◽  
Indicable Group

For a group [Formula: see text] and a set [Formula: see text], let [Formula: see text] be the monoid of all cellular automata over [Formula: see text], and let [Formula: see text] be its group of units. By establishing a characterization of surjunctive groups in terms of the monoid [Formula: see text], we prove that the rank of [Formula: see text] (i.e. the smallest cardinality of a generating set) is equal to the rank of [Formula: see text] plus the relative rank of [Formula: see text] in [Formula: see text], and that the latter is infinite when [Formula: see text] has an infinite decreasing chain of normal subgroups of finite index, condition which is satisfied, for example, for any infinite residually finite group. Moreover, when [Formula: see text] is a vector space over a field [Formula: see text], we study the monoid [Formula: see text] of all linear cellular automata over [Formula: see text] and its group of units [Formula: see text]. We show that if [Formula: see text] is an indicable group and [Formula: see text] is finite-dimensional, then [Formula: see text] is not finitely generated; however, for any finitely generated indicable group [Formula: see text], the group [Formula: see text] is finitely generated if and only if [Formula: see text] is finite.

scholarly journals On the monoid of cofinite partial isometries of $\qq{N}^n$ with the usual metric

2019 ◽  
Vol 12 (3) ◽  
pp. 51-68
Author(s):  
Oleg Gutik ◽  
Anatolii Savchuk
Keyword(s):  
Positive Integer ◽  
Symmetric Group ◽  
Natural Partial Order ◽  
Group Congruence ◽  
Partial Isometries ◽  
Group Of Units ◽  
Minimum Group ◽  
Positive Integers ◽  
Quotient Semigroup ◽  
Free Semilattice

In this paper we study the structure of the monoid Iℕn ∞ of  cofinite partial isometries of the n-th power of the set of positive integers ℕ with the usual metric for a positive integer n > 2. We describe the group of units and the subset of idempotents of the semigroup Iℕn ∞, the natural partial order and Green's relations on Iℕn ∞. In particular we show that the quotient semigroup Iℕn ∞/Cmg, where Cmg is the minimum group congruence on Iℕn ∞, is isomorphic to the symmetric group Sn and D = J in Iℕn ∞. Also, we prove that for any integer n ≥2 the semigroup Iℕn ∞  is isomorphic to the semidirect product Sn ×h(P∞(Nn); U) of the free semilattice with the unit (P∞(Nn); U)  by the symmetric group Sn.

Twisted Group Rings Whose Units Form an FC-Group

1995 ◽  
Vol 47 (2) ◽  
pp. 274-289
Author(s):  
Victor Bovdi
Keyword(s):  
Group Algebra ◽  
Group Algebras ◽  
Group Rings ◽  
Twisted Group Algebra ◽  
Group Of Units ◽  
Twisted Group

AbstractLet U(KλG) be the group of units of the infinite twisted group algebra KλG over a field K. We describe the FC-centre ΔU of U(KλG) and give a characterization of the groups G and fields K for which U(KλG) = ΔU. In the case of group algebras we obtain the Cliff-Sehgal-Zassenhaus theorem.

scholarly journals A Combinatorial Note for Harmonic Tensors

2012 ◽  
Vol 2012 ◽  
pp. 1-7
Author(s):  
Zhankui Xiao
Keyword(s):  
Symmetric Group ◽  
Group Algebra

We give another characterization of the annihilator of the space of (dual) harmonic tensors in the group algebra of symmetric group.

On p-quotients and star diagrams of the symmetric group

1953 ◽  
Vol 49 (1) ◽  
pp. 157-160 ◽  
Author(s):  
H. Farahat
Keyword(s):  
Symmetric Group ◽  
Modular Representations

The star diagram of a diagram [λ] was first obtained by Robinson (5) who denned and studied it, thus giving a unique characterization of the hook structure of [λ]. Staal (6) continued Robinson's work and obtained some important results. Recently, Littlewood (2) published a paper on the modular representations of the symmetric group in which he defined ‘congruent partitions’, ‘p-quotients’, etc., relating to the diagram [λ]. Littlewood's ideas, though superficially different from those of Robinson and Staal, are in essence equivalent to them.

scholarly journals Unit regular inverse monoids and Cliford monoids

2018 ◽  
Vol 7 (2.13) ◽  
pp. 306
Author(s):  
Sreeja V K
Keyword(s):  
Inverse Semigroup ◽  
Regular Semigroup ◽  
Group Element ◽  
Inverse Monoid ◽  
Completely Regular Semigroup ◽  
Clifford Semigroup ◽  
Inverse Monoids ◽  
Completely Regular ◽  
Group Of Units

Let S be a unit regular semigroup with group of units G = G(S) and semilattice of idempotents E = E(S). Then for every there is a such that Then both xu and ux are idempotents and we can write or .Thus every element of a unit regular inverse monoid is a product of a group element and an idempotent. It is evident that every L-class and every R-class contains exactly one idempotent where L and R are two of Greens relations. Since inverse monoids are R unipotent, every element of a unit regular inverse monoid can be written as s = eu where the idempotent part e is unique and u is a unit. A completely regular semigroup is a semigroup in which every element is in some subgroup of the semigroup. A Clifford semigroup is a completely regular inverse semigroup. Characterization of unit regular inverse monoids in terms of the group of units and the semilattice of idempotents is a problem often attempted and in this direction we have studied the structure of unit regular inverse monoids and Clifford monoids. 

scholarly journals A new characterization of symmetric group by NSE

2017 ◽  
Vol 67 (2) ◽  
pp. 427-437
Author(s):  
Azam Babai ◽  
Zeinab Akhlaghi
Keyword(s):  
Symmetric Group

scholarly journals Sn-normal semigroups

1994 ◽  
Vol 37 (3) ◽  
pp. 471-476 ◽  
Author(s):  
I. Levi ◽  
R. B. McFadden
Keyword(s):  
Symmetric Group ◽  
Transformation Semigroup ◽  
Normal Closure ◽  
Full Transformation ◽  
Group Of Units ◽  
Full Transformation Semigroup ◽  
Finite Set ◽  
Normal Ideals

Certain subsemigroups of the full transformation semigroup Tn on a finite set of cardinality n are investigated, namely those subsemigroups S of Tn, which are normalised by the symmetric group on n elements, the group of units of Tn. The Sn-normal closure of an element of Tn is determined, and the structure of the Sn-normal ideals consisting of the members of Tn whose image contains at most r elements is studied.

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