Vol. 47, No. 1, 1973

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On the number of polynomials of an idempotent algebra. II

George Grätzer and J. Płonka

Vol. 47 (1973), No. 1, 99–113
Abstract

In part I of this paper a conjecture was formulated according to which, with a few obvious exceptions, the sequence pn(U)of an idempotent algebra is eventually strictly increasing. In this paper this conjecture is verified for idempotent algebras satisfying p2(U) = 0,pa(U) > 0, and p4(U) > 0. In fact, somewhat more is proved:

Theorem. Let U be an idempotent algebra with no essentially binary polynomial and with essentially ternary and quaternary polynomials. Then the sequence

p3(U ),p4(U ),⋅⋅⋅ ,pn(U ),⋅⋅⋅

is strictly increasing, that is, for all n 2

pn(U )+ 1 ≦ pn+1(U ).

Mathematical Subject Classification
Primary: 08A25
Milestones
Received: 6 May 1972
Published: 1 July 1973
Authors
George Grätzer
J. Płonka