The Menger algebras of 2-place functions in the 2-valued logic

A Menger algebra of 2-place functions over a set Δ is (cf. [5]) a set 6 of functions mapping Δ x Δ into Δ, which is closed with respect to substitution or composition, i.e., has the property that, for any three functions F, G, H belonging to 6, the composite function F{G,H) belongs to 6. Here, F{G,H) is the function assuming the value F(G(x,y), H(x9y)) for each (x,y) in Δ x Δ. The purpose of this paper is to list all the Menger algebras of 2-place functions over {0,1}. The correspondence between these functions and the binary operators of the 2-valued logic is obvious. We denote the 16 functions (cf. [1], [2]) by