Factor-functions on partially ordered sets and reduced incidence algebras (Q2747696)

To the student of BCK-(related)-algebras, the definition of \(w\)-posets \((P,w)\) is familiar in flavor and can be viewed as a local version where uniqueness of minimal elements is not a requirement. Noting this, observations in the paper provide access to a representation theory of BCK-algebras along lines discussed here. In any case the approach taken follows (by now) classical lines in the combinatorial setting as is made clear in relevant examples. The trick is to define the \(w\)-algebra for \((P,w)\)-posets (which are locally finite) and commutative rings with 1 in the standard way with an adjustment in the convolution formula \((fg)(x)= \sum u\leq x f(u) g(w(x,u))\), to obtain the conclusion that such algebras are isomorphic to subalgebras of incidence algebras \(I(P,R)\) of \(P\) over \(R\), and that in special cases these algebras are seen to be homomorphic images of power-series rings in non-commuting variables, thus generalizing a variety of known cases from the literature significantly.