TY - JOUR
T1 - Degree 2 transformation semigroups as continuous maps on graphs
T2 - Foundations and structure
AU - Margolis, Stuart
AU - Rhodes, John
N1 - Publisher Copyright:
© 2021 World Scientific Publishing Company.
PY - 2021/9
Y1 - 2021/9
N2 - We develop the theory of transformation semigroups that have degree 2, that is, act by partial functions on a finite set such that the inverse image of points have at most two elements. We show that the graph of fibers of such an action gives a deep connection between semigroup theory and graph theory. It is known that the Krohn-Rhodes complexity of a degree 2 action is at most 2. We show that the monoid of continuous maps on a graph is the translational hull of an appropriate 0-simple semigroup. We show how group mapping semigroups can be considered as regular covers of their right letter mapping image and relate this to their graph of fibers.
AB - We develop the theory of transformation semigroups that have degree 2, that is, act by partial functions on a finite set such that the inverse image of points have at most two elements. We show that the graph of fibers of such an action gives a deep connection between semigroup theory and graph theory. It is known that the Krohn-Rhodes complexity of a degree 2 action is at most 2. We show that the monoid of continuous maps on a graph is the translational hull of an appropriate 0-simple semigroup. We show how group mapping semigroups can be considered as regular covers of their right letter mapping image and relate this to their graph of fibers.
KW - Degree of a transformation semigroup
KW - complexity of semigroups
UR - http://www.scopus.com/inward/record.url?scp=85107802257&partnerID=8YFLogxK
U2 - 10.1142/s0218196721400051
DO - 10.1142/s0218196721400051
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AN - SCOPUS:85107802257
SN - 0218-1967
VL - 31
SP - 1065
EP - 1091
JO - International Journal of Algebra and Computation
JF - International Journal of Algebra and Computation
IS - 6
ER -