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Abstract. We investigate partial monoid actions, in the sense of Megrelishvili and Schröder [12]. These are equivalent to a class of premorphisms, which we call strong premorphisms. We describe two distinct methods for constructing a monoid action from a partial monoid action: the expansion method provides a generalisation of a result of Kellendonk and Lawson [10] in the group case, whilst the approach via globalisation extends results of both [12] and [10].

We give a complete description of Green's D relation for the multiplicative semigroup of all n × n tropical matrices. Our main tool is a new variant on the duality between the row and column space of a tropical matrix (studied by Cohen, Gaubert and Quadrat and separately by Develin and Sturmfels). Unlike the existing duality theorems, our version admits a converse, and hence gives a necessary and sufficient condition for two tropical convex sets to be the row and column space of a matrix. We also show that the matrix duality map induces an isometry (with respect to the Hilbert projective metric) between the projective row space and projective column space of any tropical matrix, and establish some foundational results about Green's other relations.Tropical algebra (also known as max-plus algebra or max-algebra) is the algebra of the real numbers (sometimes augmented with −∞ and/or +∞) when equipped with the binary operations of addition and maximum. It has traditional applications in a wide range of subjects, such as combinatorial optimisation and scheduling problems [6], analysis of discrete event systems [20], control theory [9], formal language and automata theory [27,29] and combinatorial/geometric group theory [3]. More recently, exciting connections have emerged with algebraic geometry [2,25,28]; these have also led to new applications in areas such as phylogenetics [15] and statistical inference [26]. The first detailed axiomatic study was conducted by Cuninghame-Green [12] and this theory has been developed further by a number of researchers (see [1,21] for surveys).Since many of the problems which arise in application areas are naturally expressed in terms of (max-plus) linear equations, much of tropical algebra is concerned with matrices. Many researchers have had cause to prove ad hoc results about the multiplication of tropical matrices; there has also been considerable attention paid to certain special questions such as Burnside-type problems for semigroups of tropical matrices [13,17,27,29]. Surprisingly, though, there has been relatively little systematic study of these semigroups, and little is known about their abstract algebraic structure. In particular, there has been until recently no understanding of the semigroup of all matrices of a given size over the tropical semiring, comparable with the classical theory of the general linear group or full matrix semigroup over a field. The detailed study of this 1 Christopher Hollings' current address:

We introduce partial actions of weakly left E-ample semigroups, thus extending both the notion of partial actions of inverse semigroups and that of partial actions of monoids. Weakly left E-ample semigroups arise very naturally as subsemigroups of partial transformation semigroups which are closed under the unary operation α → α + , where α + is the identity map on the domain of α. We investigate the construction of 'actions' from such partial actions, making a connection with the FA-morphisms of Gomes. We observe that if the methods introduced in the monoid case by Megrelishvili and Schröder, and by the second author, are to be extended appropriately to the case of weakly left E-ample semigroups, then we must construct not global actions, but so-called incomplete actions. In particular, we show that a partial action of a weakly left E-ample semigroup is the restriction of an incomplete action. We specialize our approach to obtain corresponding results for inverse semigroups.2000 Mathematics subject classification: primary 20M30; secondary 20M18.

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