Geometric Ramifications of Invariant Expressions in the Ternary Hypercommutative Variety

Abstract

The Ternary Hypercommutative (TH) variety is an equational class of algebras determined by the identities:

• (i) [a,b,c] is invariant

• (ii) [a,b,[a,b,c]] = c

• (iii) [[a,b,c],[d,e,f],[g,h,j]] is invariant.

A word in this variety is a properly parenthetized expression consisting of variables and the ternary operation [ , , ]. If T is a TH algebra, and elements from T are substituted for the variables in a word, then the resulting element of T is the value of the word with respect to T. A word is monotone if no variable is repeated, and a monotone word is invariant if its value with respect to any TH algebra remains unchanged under any permutation of its variables. Thus to say [a,b,c] is invariant and is a shorthand way to express the identities [a,b,c] = [a,c,b] = [b,a,c] = [b,c,a] = [c,a,b] = [c,b,a]. The geometric significance of the TH variety is found in the study of intersection properties of circles with certain types of algebraic curves. For example, if α is an ellipse and a, b, c are any three points of α, then the circle (a,b,c) meets α in a unique fourth point, counting multiplicities, which we denote by our ternary operation [a,b,c]. Identities (i), (ii) are clearly satisfied by this operation, and in Fletcher (Group Circle Systems on Conics, New Frontiers of Multidisciplinary Research in STEAM-H, Springer International Publishing, Cham, 2014) it is shown that axiom (iii) also holds. We call algebraic curves whose intersection with circles satisfies the TH axioms supercyclic. We characterize supercyclic curves and determine the invariant words in the TH variety. Evidence is also given for a Conjecture regarding the significance of these words with respect to intersection properties of supercyclic curves and general algebraic curves.