Endomorphisms of semigroups of order-preserving partial transformations

Endomorphisms of semigroups of monotone transformations

“…We begin this section by recalling the following results by Fernandes et al [20] and Fernandes and Santos [23] above mentioned.…”

“…Regarding semigroups of order-preserving transformations, in 1962 Aȋzenštat [1] gave a presentation for O n from which it can be deduced that O n has only one non-trivial automorphism, for n 2. More recently, in 2010 Fernandes et al [20] found a description of the endomorphisms of O n and in 2019 Fernandes and Santos [23] determined the endomorphisms of POI n and of PO n . Descriptions of automorphisms of semigroups of order-preserving transformations and of some of their extensions, such as semigroups of monotone transformations or semigroups of orientation-preserving/reversing transformations, can be found in [3].…”

“…Theorem 1.1] and[23, Theorem 3.3]). For n 2, let T ∈ {O n , POI n , PO n } and φ : T −→ T be any mapping.…”

“…Therefore, jh −1 ⊆ Dom(f ), where j is the fixed point of h (and the image of f ). Moreover, for each pair (see [23,Lemma 3.6]). Therefore, being…”

Endomorphisms of semigroups of oriented transformations

“…For instance, this was done by Schein and Teclezghi [25,26,27] for I n in 1997 and for T n and PT n in 1998, and by Mazorchuk [23] for the Brauer-type semigroups in 2002. More recently, Fernandes et al [14] found a description of the endomorphisms of O n , Fernandes and Santos [17] determined the endomorphisms of POI n and PO n and Fernandes and Li [15] gave a characterization of the endomorphisms of OD n , PODI n and POD n . Descriptions of automorphisms of semigroups of order-preserving transformations and of some of their extensions, such as semigroups of monotone transformations or semigroups of oriented transformations, can be found in [1].…”