Local automorphisms of finitary incidence algebras

Abstract

Abstract. Let R be a commutative, indecomposable ring with identity and (P, ≤) a partially ordered set. Let F I(P ) denote the finitary incidence algebra of (P, ≤) over R. We will show that, in most cases, local automorphisms of F I(P ) are actually R-algebra automorphisms. In fact, the existence of local automorphisms which fail to be R-algebra automorphisms will depend on the chosen model of set theory and will require the existence of measurable cardinals. We will discuss local automorphisms of cartesian products as a special case in preparation of the general result.

On Local Automorphisms of sl2

“…Moreover, it is straightforward that the composition of two local automorphisms is a local automorphism and LAut(g) is a monoid. There is a statement [4, Lemma 4] that LAut(g) is a group under the usual composition, which is not true since local automorphisms are generally speaking not surjective (see Theorem 3.11 of [6]). However, the proof given in [4] works if g is finitedimensional.…”

Higher Derivations of Finitary Incidence Algebras

“…Now ([r n , n] −1 ) in = 0, except for the case i n = 0 in view of (8). The rest of the proof follows as in (i).…”

“…The study of algebraic mappings on incidence algebras was initiated by Stanley [37]. Since then, automorphisms, involutions, derivations (and their generalizations) on incidence algebras have been actively investigated, see [1,33,7,5,6,22,34,17,18,10,40,20,41,21,2,8] and the references therein.…”

Local derivations of finitary incidence algebras

“…In this short note, which was inspired by the recent preprint [10] by Courtemanche, Dugas and Herden, we adapt the ideas from [21] to the infinite case using the technique elaborated in [14,16]. More precisely, we show that each R-linear local derivation of the finitary incidence algebra F I(P, R) of an arbitrary poset P over a commutative unital ring R is a derivation, giving thus another partial generalization of [21, Theorem 3].…”