A Short Proof of Representability of Fork Algebras

In this paper a strong relation is demonstrated between fork algebras and quasi-projective relation algebras. With the help of the representation theorem of quasi-projective relation algebras, a short proof is given for the representation theorem of fork algebras. Fork algebras, due to their expressive power and applicability in computing science, have been intensively studied in the last four years. Their literature is alive and productive. See e. in algebra, there are two kinds of representation theorems: strong and weak representation. For fork algebras strong representation was proved impossible in 15], 17], 16]. However, weak representation is still possible and is quite useful (e.g. 1]). The distinction between the two kinds of representation is not absolutely necessary for understanding the main contribution of this paper, therefore we postpone the description of this distinction to remark 0.10 at the end of the paper. In the main bulk of the paper we concentrate on weak representation. Hence for brevity, we often write representation instead of weak representation, hoping that context will help. Several papers concentrate on giving a proof or an outline of proof for weak repre-sentability of fork algebras (e.g. 1], 2], 8]). The contribution of the present note is threefold. (i) We provide a very short and easy proof for this representation theorem. (ii) We generalize the representation theorem to a much broader class called pre-fork algebras from the originally discussed narrower class in 1]. Originally, fork algebras did not form an equational class and their axiomatization involved a quite complicated formula not equivalent to any equation or even quasi-equation. The new pre-fork algebras admit a simple equational axiomatization, and contain all thèold' fork algebras. It seems that in their latest paper Baum, Frias, Haeberer and LL opez adopt the wider class we call here pre-fork algebras as the central topic of study cf. 1]. (iii) We elaborate the connection between the recent theory of fork algebras and Tarski's theory of pairing relation algebras which is a classical branch of algebraic logic going back to 1949 20, p. 168]. For historical reasons we will use the 791