9 research outputs found
On semigroups of endomorphisms of a chain with restricted range
Let be a finite or infinite chain and let be the monoid of all
endomorphisms of . In this paper, we describe the largest regular
subsemigroup of and Green's relations on . In fact, more
generally, if is a nonempty subset of and the subsemigroup of
of all elements with range contained in , we characterize the largest
regular subsemigroup of and Green's relations on . Moreover,
for finite chains, we determine when two semigroups of the type are
isomorphic and calculate their ranks.Comment: To appear in Semigroup Foru
Natural Partial Orders on Transformation Semigroups with Fixed Sets
Let X be a nonempty set. For a fixed subset Y of X, let FixX,Y be the set of all self-maps on X which fix all elements in Y. Then FixX,Y is a regular monoid under the composition of maps. In this paper, we characterize the natural partial order on Fix(X,Y) and this result extends the result due to Kowol and Mitsch. Further, we find elements which are compatible and describe minimal and maximal elements
Sandwich semigroups in locally small categories I: Foundations
Fix (not necessarily distinct) objects and of a locally small
category , and write for the set of all morphisms . Fix a
morphism , and define an operation on by
for all . Then is a
semigroup, known as a sandwich semigroup, and denoted by . This
article develops a general theory of sandwich semigroups in locally small
categories. We begin with structural issues such as regularity, Green's
relations and stability, focusing on the relationships between these properties
on and the whole category . We then identify a natural condition
on , called sandwich regularity, under which the set Reg of all
regular elements of is a subsemigroup of . Under this
condition, we carefully analyse the structure of the semigroup Reg,
relating it via pullback products to certain regular subsemigroups of
and , and to a certain regular sandwich monoid defined on a subset of
; among other things, this allows us to also describe the
idempotent-generated subsemigroup of . We also
study combinatorial invariants such as the rank (minimal size of a generating
set) of the semigroups , Reg and ;
we give lower bounds for these ranks, and in the case of Reg and
show that the bounds are sharp under a certain condition
we call MI-domination. Applications to concrete categories of transformations
and partial transformations are given in Part II.Comment: 23 pages, 1 figure. V2: updated according to referee report, expanded
abstract, to appear in Algebra Universali
Sandwich semigroups in locally small categories II: Transformations
Fix sets and , and write for the set of all
partial functions . Fix a partial function , and define the
operation on by for
. The sandwich semigroup
is denoted . We apply general results from Part I to
thoroughly describe the structural and combinatorial properties of
, as well as its regular and idempotent-generated
subsemigroups, Reg and .
After describing regularity, stability and Green's relations and preorders, we
exhibit Reg as a pullback product of certain regular
subsemigroups of the (non-sandwich) partial transformation semigroups
and , and as a kind of "inflation" of
, where is the image of the sandwich element . We also
calculate the rank (minimal size of a generating set) and, where appropriate,
the idempotent rank (minimal size of an idempotent generating set) of
, Reg and . The same program is also carried out for sandwich
semigroups of totally defined functions and for injective partial functions.
Several corollaries are obtained for various (non-sandwich) semigroups of
(partial) transformations with restricted image, domain and/or kernel.Comment: 35 pages, 11 figures, 1 table. V2: updated according to referee
report, expanded abstract, to appear in Algebra Universali
Semigroups of transformations with fixed sets
Click on the link to view the abstract.Keywords: Transformation semigroup, Green's relations, ideal, rankQuaestiones Mathematicae 36(2013), 79-9
Greenâs Relations on a Semigroup of Transformations with Restricted Range that Preserves an Equivalence Relation and a Cross-Section
Let T(X,Y) be the semigroup consisting of all total transformations from X into a fixed nonempty subset Y of X. For an equivalence relation ρ on X, let ρ^ be the restriction of ρ on Y, R a cross-section of Y/ρ^ and define T(X,Y,ρ,R) to be the set of all total transformations α from X into Y such that α preserves both ρ (if (a,b)∈ρ, then (aα,bα)∈ρ) and R (if r∈R, then rα∈R). T(X,Y,ρ,R) is then a subsemigroup of T(X,Y). In this paper, we give descriptions of Green’s relations on T(X,Y,ρ,R), and these results extend the results on T(X,Y) and T(X,ρ,R) when taking ρ to be the identity relation and Y=X, respectively
On Semigroups of Orientation-preserving Transformations with Restricted Range
Let Xn be a chain with n elements (n â â), and let n be the monoid of all orientation-preserving transformations of Xn. In this article, for any nonempty subset Y of Xn, we consider the subsemigroup n(Y) of n of all transformations with range contained in Y: We describe the largest regular subsemigroup of n(Y), which actually coincides with its subset of all regular elements. Also, we determine when two semigroups of the type n(Y) are isomorphic and calculate their ranks. Moreover, a parallel study is presented for the correspondent subsemigroups of the monoid ân of all either orientation-preserving or orientation-reversing transformations of Xn.info:eu-repo/semantics/publishedVersio