The saga of the High School Identities

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.

Section: Wilkie's Identity In Natural Hsi-algebrasmentioning

confidence: 99%

“…The smallest countermodel known so far is due to Burris and Yeats [5] (2004) and it has 12 elements.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Asatryan¹

2008

“…Burris and Lee [2] and later Jackson [6] constructed new such HSI-algebras (counterexamples to Wilkie's identity) with smaller number of elements. The smallest counterexample was constructed by Burris and Yeats [3].…”

Section: Introductionmentioning

confidence: 99%

A solution to identities problem in 2‐element HSI‐algebras

Asatryan

2004

An intuitionistic formula hierarchy based on high‐school identities

Brock-Nannestad

Ilik

2019

To the memory of Kosta Došen and Roy DyckhoffWe revisit the notion of intuitionistic equivalence and formal proof representations by adopting the view of formulas as exponential polynomials. After observing that most of the invertible proof rules of intuitionistic (minimal) propositional sequent calculi are formula (i.e., sequent) isomorphisms corresponding to the highschool identities, we show that one can obtain a more compact variant of a proof system, consisting of noninvertible proof rules only, and where the invertible proof rules have been replaced by a formula normalization procedure. Moreover, for certain proof systems such as the G4ip sequent calculus of Vorob'ev, Hudelmaier, and Dyckhoff, it is even possible to see all of the non-invertible proof rules as strict inequalities between exponential polynomials; a careful combinatorial treatment is given in order to establish this fact. Finally, we extend the exponential polynomial analogy to the first-order quantifiers, showing that it gives rise to an intuitionistic hierarchy of formulas, resembling the classical arithmetical hierarchy, and the first one that classifies formulas while preserving isomorphism.