We study the relations between partial and global group cohomology. We show, in particular, that given a unital partial action of a group G on a ring A, such that A is a direct product of indecomposable rings, then any partial n-cocycle with values in A is globalizable.

Let X X be a finite connected poset and K K a field. We give a full description of the Lie automorphisms of the incidence algebra I ( X , K ) I(X,K) . In particular, we show that they are in general not proper.

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We introduce the concept of an extension of a semilattice of groups A by a group G and describe all the extensions of this type which are equivalent to the crossed products A * Θ G by twisted partial actions Θ of G on A. As a consequence, we establish a one-to-one correspondence, up to an isomorphism, between twisted partial actions of groups on semilattices of groups and socalled Sieben twisted modules over E-unitary inverse semigroups.1 2 MIKHAILO DOKUCHAEV AND MYKOLA KHRYPCHENKO assume that A is inverse, i. e. A is a semilattice of abelian groups. Thus, when passing from partial actions of G to actions of S(G), we obtain a relation between partial G-modules and modules over inverse semigroups, and this way the cohomology of inverse semigroups comes into the picture. The latter was introduced in the pioneering paper by H. Lausch [26], with a more general approach given by M. Loganathan in [31], where H. Lausch's cohomology was derived as the cohomology of a suitable small category. Moreover, in the context of regular monoids M. Loganathan's cohomology can be seen as the cohomology of J. Leech [30] (see [31, Theorem 2.9]). M. Loganathan's view, amongst other advantages, allows one to use general facts from categories and homological algebra, in particular, when dealing with functorial properties of cohomology groups. Nevertheless, it turned out that the category of partial G-modules is not abelian, so that technically it is more appropriate for us to deal with the down-to-earth H. Lausch's method. In fact, our category is made up of abelian "pieces", each of which is isomorphic to a category of modules over some inverse semigroup. These inverse semigroups run over those homomorphic images of S(G), which are the max-generated F -inverse monoids S (see [28, p. 196]) whose maximum group image is G. For these monoids [15] gives a rather comprehensive connection between the Lausch-Leech-Loganathan cohomology of S and the partial cohomology of G. Moreover, it was also clarified in [15] that the partial Schur multiplier of a group G, which appeared in the theory of partial projective group representations, is a disjoint union of second cohomology groups with values in partial (in general, non-trivial) G-modules. Influenced by H. Lausch's construction of a free module [26], it was possible to define free partial G-modules and obtain the partial cohomology groups via free resolutions [15].The present paper is stimulated by the desire to interpret the second partial cohomology groups in terms of extensions. In doing this it became clear, at some point, that we may abandon the restriction on a partial action to be unital, imposed by cohomology theory in [15], covering thus a more general situation. The starting point is the concept of an extension of a semilattice of groups A by a group G (see Section 4) and the key example of an extension is the crossed product A * Θ G by a twisted partial action Θ of G on A (Section 5). These are related to the notion of an extension of A by an inverse semigroup S

Abstract. Let X be a locally finite preordered set, R a commutative ring with identity and I(X, R) the incidence algebra of X over R. In this note we prove that each Lie derivation of I(X, R) is proper, provided that R is 2-torsion free.