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  • PARTIAL ORDERS ON SEMIGROUPS OF PARTIAL TRANSFORMATIONS WITH RESTRICTED RANGE

  • Part of
  • KRITSADA SANGKHANAN, JINTANA SANWONG
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 86 / Issue 1 / August 2012
    Published online by Cambridge University Press:
    16 February 2012, pp. 100-118
    Print publication:
    August 2012
    • Article
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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.