×

On the role of the polynomial (X\(\rightarrow Y)\rightarrow Y\) in some implicative algebras. (English) Zbl 0621.03043

The main result of the paper gives a relation, by means of the associated partial order to any \(BCK^ *\)-algebra (dual to BCK-algebra), between the interpretations of the polynomial (x\(\to y)\to y\). Concretely, if p(x,y) is a two-ary polynomial symbol and A is a \(BCK^ *\)-algebra, then for any a,b\(\in A\), \(a,b\leq p(a,b)^ A\) iff (a\(\to b)\to b\leq p(a,b)^ A\) or (b\(\to a)\to a\leq p(a,b)^ A\). Using this result it is shown that the only way to give the join, if it exists, by means of a two-ary polynomial in a \(BCK^ *\)-algebra (and hence in a Hilbert algebra) is by using the polynomial symbol (x\(\to y)\to y\).

MSC:

03G25 Other algebras related to logic
PDFBibTeX XMLCite
Full Text: DOI