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 -