TY - JOUR
T1 - Sandwich semigroups in locally small categories I : foundations
AU - Dolinka, Igor
AU - Đurđev, Ivana
AU - East, James
AU - Honyam, Preeyanuch
AU - Sangkhanan, Kritsada
AU - Sanwong, Jintana
AU - Sommanee, Worachead
PY - 2018
Y1 - 2018
N2 - Fix (not necessarily distinct) objects i and j of a locally small category S, and write Sij for the set of all morphisms i→ j. Fix a morphism a ∈ Sji, and define an operation ⋆ a on Sij by x⋆a y= xay for all x, y ∈ Sij. Then (Sij, ⋆a) is a semigroup, known as a sandwich semigroup, and denoted by Sija. 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 Sija and the whole category S. We then identify a natural condition on a, called sandwich regularity, under which the set Reg(Sija) of all regular elements of Sija is a subsemigroup ofà Sija. Under this condition, we carefully analyse the structure of the semigroup Reg(Sija), relating it via pullback products to certain regular subsemigroups of Sii and Sjj, and to a certain regular sandwich monoid defined on a subset of Sji; among other things, this allows us to also describe the idempotent-generated subsemigroup E(Sija) of Sija. We also study combinatorial invariants such as the rank (minimal size of a generating set) of the semigroups Sija, Reg(Sija) and E(Sija); we give lower bounds for these ranks, and in the case of Reg(Sija) and E(Sija) 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. NOTE: SOME OF THE SCIENTIC SYMBOLS CAN NOT BE REPRESENTED CORRECTLY IN THE ABSTRACT. PLEASE READ WITH CAUTION AND REFER TO THE ORIGINAL THESIS.
AB - Fix (not necessarily distinct) objects i and j of a locally small category S, and write Sij for the set of all morphisms i→ j. Fix a morphism a ∈ Sji, and define an operation ⋆ a on Sij by x⋆a y= xay for all x, y ∈ Sij. Then (Sij, ⋆a) is a semigroup, known as a sandwich semigroup, and denoted by Sija. 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 Sija and the whole category S. We then identify a natural condition on a, called sandwich regularity, under which the set Reg(Sija) of all regular elements of Sija is a subsemigroup ofà Sija. Under this condition, we carefully analyse the structure of the semigroup Reg(Sija), relating it via pullback products to certain regular subsemigroups of Sii and Sjj, and to a certain regular sandwich monoid defined on a subset of Sji; among other things, this allows us to also describe the idempotent-generated subsemigroup E(Sija) of Sija. We also study combinatorial invariants such as the rank (minimal size of a generating set) of the semigroups Sija, Reg(Sija) and E(Sija); we give lower bounds for these ranks, and in the case of Reg(Sija) and E(Sija) 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. NOTE: SOME OF THE SCIENTIC SYMBOLS CAN NOT BE REPRESENTED CORRECTLY IN THE ABSTRACT. PLEASE READ WITH CAUTION AND REFER TO THE ORIGINAL THESIS.
KW - categories (mathematics)
KW - idempotents
KW - semigroup algebras
KW - transformations (mathematics)
UR - http://handle.westernsydney.edu.au:8081/1959.7/uws:49995
U2 - 10.1007/s00012-018-0537-5
DO - 10.1007/s00012-018-0537-5
M3 - Article
SN - 0002-5240
VL - 79
JO - Algebra Universalis
JF - Algebra Universalis
IS - 3
M1 - 75
ER -