Multiplicatively idempotent HSI algebras satisfy all equations of ℕ
Abstract
An algebra with binary operations +,·,↑ and constant $1$ is called an HSI algebra if it satisfies the basic commutative semiring laws for +,·,1 on~ℕ as well as the familiar index laws for exponentiation ↑. These basic axioms, known as the ``High School Identities'' are known to be incomplete, and an algebra satisfying HSI but failing an equation valid on bf N:=⟨ ℕ;+,·,↑,1⟩ is called a \emph{Gurevič algebra}. It is currently unknown if there is an algorithm to recognise finite Gurevič algebras, and the best current result is that there exists a 12-element Gurevič algebra, and that none of the five 2-element HSI algebras are Gurevič algebras. We explain how the work of Alex Wilkie can be used to provide an algorithm for deciding validity of the equational laws of ⟨ ℕ;+,·,↑,1⟩, and use this to show that multiplicatively-idempotent HSI algebras satisfy all valid laws of ⟨ ℕ;+,·,↑,1⟩. As consequences of this result, we show that all 44 HSI algebras on 3 elements lie in the variety of ⟨ ℕ;+,·,↑,1⟩ (that is, are not Gurevič algebras), as well as 597 of the 657 models on 4 elements and 11158 of the 13577 models on 5 elements. A further consequence is that every Brouwerian lattice lies in the variety of ⟨ ℕ;+,·,↑,1⟩ (up to a simple term equivalence), showing that the variety of ⟨ ℕ;+,·,↑,1⟩ has continuum many subvarieties. The constant-free signature is also explored, and it is shown that in this case all $2$-element models of the constant-free High School Laws satisfy all valid constant-free laws in ⟨ ℕ;+,·,↑⟩.
Related
- Congruence-simple multiplicatively idempotent semirings and multiplicative divisibility
- The Finite Basis Problem for Multiplicatively Idempotent Semirings
- Idempotent-free non-solvable evolution algebras over ℂ
- Idempotent words in finite-dimensional simple Jordan algebras
- Hyperbolic groups satisfy the Boone–Higman conjecture
BibTeX
@misc{alsulami2026multiplicatively,
title = {{Multiplicatively idempotent HSI algebras satisfy all equations of $\mathbbN$}},
author = {Tumadhir Alsulami and Marcel Jackson and Michael Kinyon},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16922},
eprint = {2609.16922},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16922},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Multiplicatively idempotent HSI algebras satisfy all equations of ℕ
AU - Tumadhir Alsulami
AU - Marcel Jackson
AU - Michael Kinyon
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16922
UR - https://arxiv.org/abs/2609.16922
AB - An algebra with binary operations +,·,↑ and constant $1$ is called an HSI algebra if it satisfies the basic commutative semiring laws for +,·,1 on~ℕ as well as the familiar index laws for exponentiation ↑. These basic axioms, known as the ``High School Identities'' are known to be incomplete, and an algebra satisfying HSI but failing an equation valid on bf N:=⟨ ℕ;+,·,↑,1⟩ is called a \emph{Gurevič algebra}. It is currently unknown if there is an algorithm to recognise finite Gurevič algebras, and the best current result is that there exists a 12-element Gurevič algebra, and that none of the five 2-element HSI algebras are Gurevič algebras. We explain how the work of Alex Wilkie can be used to provide an algorithm for deciding validity of the equational laws of ⟨ ℕ;+,·,↑,1⟩, and use this to show that multiplicatively-idempotent HSI algebras satisfy all valid laws of ⟨ ℕ;+,·,↑,1⟩. As consequences of this result, we show that all 44 HSI algebras on 3 elements lie in the variety of ⟨ ℕ;+,·,↑,1⟩ (that is, are not Gurevič algebras), as well as 597 of the 657 models on 4 elements and 11158 of the 13577 models on 5 elements. A further consequence is that every Brouwerian lattice lies in the variety of ⟨ ℕ;+,·,↑,1⟩ (up to a simple term equivalence), showing that the variety of ⟨ ℕ;+,·,↑,1⟩ has continuum many subvarieties. The constant-free signature is also explored, and it is shown that in this case all $2$-element models of the constant-free High School Laws satisfy all valid constant-free laws in ⟨ ℕ;+,·,↑⟩.
ER -