Comparison semigroups and algebras of transformations |
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Authors: | Tim Stokes |
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Institution: | 1. Department of Mathematics, The University of Waikato, Hamilton, New Zealand
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Abstract: | We characterize algebras of transformations on a set under the operations of composition and the pointwise switching function
defined as follows: (f,g)h,k](x)=h(x) if f(x)=g(x), and k(x) otherwise. The resulting algebras are both semigroups and comparison algebras in the sense of Kennison. The same characterization
holds for partial transformations under composition and a suitable generalisation of the quaternary operation in which agreement
of f,g includes cases where neither is defined. When a zero element is added (modelling the empty function), the resulting signature
is rich enough to encompass many operations on semigroups of partial transformations previously considered, including set
difference and intersection, restrictive product, and a functional analog of union. When an identity element is also added
(modelling the identity function), further domain-related operations can be captured. |
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Keywords: | |
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