TY - JOUR
T1 - Some combinatorial problems in the theory of partial transformation semigroups
AU - Umar, A.
N1 - Publisher Copyright:
© Journal Algebra and Discrete Mathematics.
PY - 2014
Y1 - 2014
N2 - Let Xn = {1, 2,..., n}. On a partial transformation α : Domα ⊆ Xn → Im α ⊆ Xn of Xn the following parameters are defined: the breadth or width of α is | Domα |, the collapse of α is c(α) =| ∪t∈Imα{tα−1 :| tα−1 |≥ 2} |, fix of α is f(α) =| {x ∈ Xn : xα = x} |, the height of α is | Im α |, and the right [left] waist of α is max(Im α) [min(Im α)]. The cardinalities of some equivalences defined by equalities of these parameters on Tn, the semigroup of full transformations of Xn, and Pn the semigroup of partial transformations of Xn and some of their notable subsemigroups that have been computed are gathered together and the open problems highlighted.
AB - Let Xn = {1, 2,..., n}. On a partial transformation α : Domα ⊆ Xn → Im α ⊆ Xn of Xn the following parameters are defined: the breadth or width of α is | Domα |, the collapse of α is c(α) =| ∪t∈Imα{tα−1 :| tα−1 |≥ 2} |, fix of α is f(α) =| {x ∈ Xn : xα = x} |, the height of α is | Im α |, and the right [left] waist of α is max(Im α) [min(Im α)]. The cardinalities of some equivalences defined by equalities of these parameters on Tn, the semigroup of full transformations of Xn, and Pn the semigroup of partial transformations of Xn and some of their notable subsemigroups that have been computed are gathered together and the open problems highlighted.
KW - Breadth
KW - Collapse
KW - Fix
KW - Full transformation
KW - Height and right (left) waist of a transformation
KW - Idempotents and nilpotents
KW - Partial transformation
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M3 - Article
AN - SCOPUS:84919731604
SN - 1726-3255
VL - 17
SP - 110
EP - 134
JO - Algebra and Discrete Mathematics
JF - Algebra and Discrete Mathematics
IS - 1
ER -