On models of exponentiation. Identities in the HSI‐algebra of posets

Abstract

We prove that Wilkie's identity holds in those natural HSI-algebras where each element has finite decomposition into components.Further, we construct a bunch of HSI-algebras that satisfy all the identities of the set of positive integers N. Then, based on the constructed algebras, we prove that the identities of N hold in the HSI-algebra of finite posets when the value of each variable is a poset having an isolated point.

Axioms and decidability for type isomorphism in the presence of sums

“…= p. This representation of the axioms is inspired from the one of Asatryan [1]. Note that, for any tz, there are infinitely many axioms having tz on the left hand side that can be used.…”

“…We thus have ∀f, g ∈ E(HSI f . 1 For various results around this problem, please look at the survey articles [2] and [3].…”

“…The left-hand side of the new axioms is always an expression of form t z , while the right hand side can either be an expression t u -whenever z and u are equal polynomials (which can be decided by bringing them into canonical form) -or an expression reflecting the structure of z when possible: when z has no negative coefficients (we use in that case letters p, q instead of z, u), we can fully reflect it into the language, that is, one can prove HSI* ⊢ t p . = p. This representation of the axioms is inspired from the one of Asatryan [19]. Note that, for any t z , there are infinitely many axioms having t z on the left hand side that can be used.…”