Affine spaces as models for regular identities (Q2773275)

Two sets of identities with no finite models were given by \textit{J. Dudek} and \textit{J. Galuszka} [Discuss. Math., Algebra Stoch. Methods 15, 101-119 (1995; Zbl 0835.08004)] and \textit{J. Dudek} and \textit{A. Kisielewicz} [Notre Dame J. Formal Logic 30, 624-628 (1989; Zbl 0694.03025)]. These consist of the type \((2,2)\) identities \( x+x=x\), \(x\circ x=x\), \(x+y=y+x\), and \((x+y)\circ z= (x+z)\circ y\) together with either \(x\circ y=y\circ x\) or \(x\circ (y+z)=y\circ (x+z)\). This paper examines the properties of the first four identities given above. It is shown that affine spaces over \(\text{GF}(p)\) for \(p\geq 5\) are proper models of these identities, and also that any proper algebra \((A,+,\circ)\) which satisfies these identities has at least 13 essentially ternary term functions. There are exactly 13 if and only if \(A\) is term-equivalent to a nontrivial affine space over GF(5).