### Is Word Order Asymmetry Mathematically Expressible?

#### Abstract

The computational procedure for human natural language (CHL) shows an asymmetry in unmarked orders for S, O, and V. Following Lyle Jenkins, it is speculated that the asymmetry is expressible as a group-theoretical factor (included in Chomskyâ€™s third factor): â€œ[W]ord order types would be the (asymmetric) stable solutions of the symmetric still-to-be-discovered â€˜equationsâ€™ governing word order distributionâ€. A possible â€œsymmetric equationâ€ is a linear transformation f(x) = y, where function f is a set of merge operations (transformations) expressed as a set of symmetric transformations of an equilateral triangle, x is the universal base vP input expressed as the identity triangle, and y is a mapped output tree expressed as an output triangle that preserves symmetry. Although the symmetric group S3 of order 3! = 6 is too simple, this very simplicity is the reason that in the present work cost differences are considered among the six symmetric operations of S3. This article attempts to pose a set of feasible questions for future research.

#### Keywords

cost; economy; equilibrium; Galois group; geometry; symmetry; third factor; transformation; unmarked word order

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