Characterizations of Cayley graphs of finite transformation semigroups with restricted range

2020 ◽  
pp. 2150041
Author(s):  
Chunya Tisklang ◽  
Sayan Panma
Keyword(s):  
Cayley Graphs ◽  
Transformation Semigroup ◽  
Restricted Range ◽  
Transformation Semigroups ◽  
Finite Transformation ◽  
Empty Subset

The transformation semigroup with restricted range [Formula: see text] is the set of all functions from a set [Formula: see text] into a non-empty subset [Formula: see text] of [Formula: see text]. In this paper, we characterize Cayley graphs of [Formula: see text] with the connection set [Formula: see text]. Moreover, the undirected property of Cayley graphs Cay [Formula: see text] is studied.

scholarly journals Characterization of finite amenable transformation semigroups

1973 ◽  
Vol 15 (1) ◽  
pp. 86-93 ◽  
Author(s):  
Carroll Wilde
Keyword(s):  
Transformation Semigroup ◽  
Sufficient Conditions ◽  
Mean Value ◽  
Explicit Description ◽  
Transformation Semigroups ◽  
Shift Operators ◽  
Finite Transformation ◽  
Invariant Means ◽  
Necessary And Sufficient

Abstract. In this paper we develop necessary and sufficient conditions for a finite transformation semigroup to have a mean value which is invariant under the induced shift operators. The structure of such transformation semigroups is described and an explicit description of all possible invariant means given.

scholarly journals On relative ranks of finite transformation semigroups with restricted range

2021 ◽  
Vol 73 (5) ◽  
pp. 617-626
Author(s):  
I. Dimitrova ◽  
J. Koppitz
Keyword(s):  
Restricted Range ◽  
Finite Chain ◽  
Transformation Semigroups ◽  
Relative Rank ◽  
Generating Sets ◽  
Finite Transformation

UDC 512.5 We determine the relative rank of the semigroup of all transformations on a finite chain with restricted range modulo the set of all orientation-preserving transformations in Moreover, we state the relative rank of the semigroup modulo the set of all order-preserving transformations in In both cases we characterize the minimal relative generating sets.  

scholarly journals The complexity of properties of transformation semigroups

2019 ◽  
Vol 30 (03) ◽  
pp. 585-606
Author(s):  
Lukas Fleischer ◽  
Trevor Jack
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
General Result ◽  
Transformation Semigroup ◽  
Transformation Semigroups ◽  
Completely Regular ◽  
Finite Transformation ◽  
Left And Right

We investigate the computational complexity for determining various properties of a finite transformation semigroup given by generators. We introduce a simple framework to describe transformation semigroup properties that are decidable in [Formula: see text]. This framework is then used to show that the problems of deciding whether a transformation semigroup is a group, commutative or a semilattice are in [Formula: see text]. Deciding whether a semigroup has a left (respectively, right) zero is shown to be [Formula: see text]-complete, as are the problems of testing whether a transformation semigroup is nilpotent, [Formula: see text]-trivial or has central idempotents. We also give [Formula: see text] algorithms for testing whether a transformation semigroup is idempotent, orthodox, completely regular, Clifford or has commuting idempotents. Some of these algorithms are direct consequences of the more general result that arbitrary fixed semigroup equations can be tested in [Formula: see text]. Moreover, we show how to compute left and right identities of a transformation semigroup in polynomial time. Finally, we show that checking whether an element is regular is [Formula: see text]-complete.

Naturally Ordered Transformation Semigroups Preserving an Equivalence Relation and a Cross-section

2011 ◽  
Vol 18 (03) ◽  
pp. 523-532 ◽  
Author(s):  
Lei Sun ◽  
Weina Deng ◽  
Huisheng Pei
Keyword(s):  
Cross Section ◽  
Equivalence Relation ◽  
Transformation Semigroup ◽  
Natural Order ◽  
Transformation Semigroups ◽  
Full Transformation ◽  
Minimal Elements ◽  
Full Transformation Semigroup

The paper is concerned with the so-called natural order on the semigroup [Formula: see text], where [Formula: see text] is the full transformation semigroup on a set X, E is a non-trivial equivalence on X and R is a cross-section of the partition X/E induced by E. We determine when two elements of TE(X,R) are related under this order, find elements of TE(X,R) which are compatible with ≤ on TE(X,R), and observe the maximal and minimal elements and the covering elements.

scholarly journals The structure of elements in finite full transformation semigroups

2005 ◽  
Vol 71 (1) ◽  
pp. 69-74 ◽  
Author(s):  
Gonca Ayik ◽  
Hayrullah Ayik ◽  
Yusuf Ünlü ◽  
John M. Howie
Keyword(s):  
Finite Semigroup ◽  
Transformation Semigroup ◽  
Transformation Semigroups ◽  
Full Transformation ◽  
Full Transformation Semigroup

The index and period of an element a of a finite semigroup are the smallest values of m ≥ 1 and r ≥ 1 such that am+r = am. An element with index m and period 1 is called an m-potent element. For an element α of a finite full transformation semigroup with index m and period r, a unique factorisation α = σβ such that Shift(σ) ∩ Shift(β) = ∅ is obtained, where σ is a permutation of order r and β is an m-potent. Some applications of this factorisation are given.

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