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PARTIAL ORDERS ON SEMIGROUPS OF PARTIAL TRANSFORMATIONS WITH RESTRICTED RANGE

Part of: Semigroups

Published online by Cambridge University Press:  16 February 2012

KRITSADA SANGKHANAN  and
JINTANA SANWONG
KRITSADA SANGKHANAN
Affiliation:
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiangmai 50200, Thailand (email: [email protected])
JINTANA SANWONG*
Affiliation:
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiangmai 50200, Thailand Material Science Research Center, Faculty of Science, Chiang Mai University, Thailand (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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Let X be any set and P(X) the set of all partial transformations defined on X, that is, all functions α:A→B where A,B are subsets of X. Then P(X) is a semigroup under composition. Let Y be a subset of X. Recently, Fernandes and Sanwong defined PT(X,Y )={α∈P(X):Xα⊆Y } and defined I(X,Y ) to be the set of all injective transformations in PT(X,Y ) . Hence PT(X,Y ) and I(X,Y ) are subsemigroups of P(X) . In this paper, we study properties of the so-called natural partial order ≤ on PT(X,Y ) and I(X,Y ) in terms of domains, images and kernels, compare ≤ with the subset order, characterise the meet and join of these two orders, then find elements of PT(X,Y ) and I(X,Y ) which are compatible. Also, the minimal and maximal elements are described.

Keywords

partial transformation semigroupsnatural ordercompatibilitymaximal and minimal elements

MSC classification

Secondary: 20M20: Semigroups of transformations, etc.
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 86 , Issue 1 , August 2012 , pp. 100 - 118
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

Footnotes

The first author thanks the Development and Promotion of Science and Technology talents project, Thailand, for its financial support. He also thanks the Graduate School, Chiang Mai University, Chiangmai, Thailand, for its financial support that he received during the preparation of this paper. The second author thanks the National Research University Project under the Office of the Higher Education Commission, Thailand, for its financial support.

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