IJPAM: Volume 80, No. 4 (2012)

SEMIGROUPS OF INJECTIVE PARTIAL LINEAR
TRANSFORMATIONS WITH RESTRICTED RANGE:
GREEN'S RELATIONS AND PARTIAL ORDERS

Kritsada Sangkhanan$^1$, Jintana Sanwong$^2$
Department of Mathematics
Faculty of Science
Chiang Mai University
Chiangmai 50200, THAILAND


Abstract. Let $V$ be any vector space and $I(V)$ the set of all partial injective linear transformations defined on $V$, that is, all injective linear transformations $\alpha:A\to B$ where $A,B$ are subspaces of $V$. Then $I(V)$ is a semigroup under composition. Let $W$ be a subspace of $V$. Define $I(V,W)=\{\alpha\in I(V):V\alpha\subseteq W\}$. So $I(V,W)$ is a subsemigroup of $I(V)$. In this paper, we present the largest regular subsemigroup of $I(V,W)$ and determine its Green's relations. Furthermore, we study the natural partial order $\leq$ on $I(V,W)$ in terms of domains and images, compare $\leq$ with the subset order and find elements of $I(V,W)$ which are compatible.

Received: August 27, 2012

AMS Subject Classification: 20M20

Key Words and Phrases: regular elements, Green's relations, injective partial linear transformation semigroups, natural order, compatibility

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 80
Issue: 4