乘法幂等HSI代数满足$\mathbb{N}$的所有方程

中文摘要 AI 辅助

一个具有二元运算$+,\cdot,\uparrow$和常数$1$的代数称为HSI代数，如果它满足$\mathbb{N}$上关于$+,\cdot,1$的基本交换半环定律以及关于幂运算$\uparrow$的常见指数定律。这些基本公理被称为“高中恒等式”，已知是不完备的，满足$\HSI$但违反在${\bf N}:=\langle \mathbb{N};+,\cdot,\uparrow,1\rangle$上有效的方程的代数称为\emph{Gurevič代数}。目前尚不清楚是否存在识别有限Gurevič代数的算法，目前最好的结果是存在一个12元素的Gurevič代数，并且五个2元素HSI代数中没有一个是Gurevič代数。我们解释了如何利用Alex Wilkie的工作来提供一种判定$\langle \mathbb{N};+,\cdot,\uparrow,1\rangle$的等式定律有效性的算法，并利用该算法证明乘法幂等HSI代数满足$\langle \mathbb{N};+,\cdot,\uparrow,1\rangle$的所有有效定律。作为该结果的结果，我们证明了3元素上的所有44个HSI代数都属于$\langle \mathbb{N};+,\cdot,\uparrow,1\rangle$的簇（即不是Gurevič代数），4元素上的657个模型中的597个以及5元素上的13577个模型中的11158个也属于该簇。进一步的结果是，每个Brouwerian格都属于$\langle \mathbb{N};+,\cdot,\uparrow,1\rangle$的簇（在简单的项等价意义下），这表明$\langle \mathbb{N};+,\cdot,\uparrow,1\rangle$的簇具有连续多个子簇。还探讨了无常数签名，并证明在这种情况下，无常数高中定律的所有2元素模型都满足$\langle \mathbb{N};+,\cdot,\uparrow\rangle$中的所有有效无常数定律。

英文摘要

An algebra with binary operations $+,\cdot,\uparrow$ and constant $1$ is called an HSI algebra if it satisfies the basic commutative semiring laws for $+,\cdot,1$ on~$\mathbb{N}$ as well as the familiar index laws for exponentiation $\uparrow$. 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 ${\bf N}:=\langle \mathbb{N};+,\cdot,\uparrow,1\rangle$ 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 $\langle \mathbb{N};+,\cdot,\uparrow,1\rangle$, and use this to show that multiplicatively-idempotent HSI algebras satisfy all valid laws of $\langle \mathbb{N};+,\cdot,\uparrow,1\rangle$. As consequences of this result, we show that all 44 HSI algebras on 3 elements lie in the variety of $\langle \mathbb{N};+,\cdot,\uparrow,1\rangle$ (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 $\langle \mathbb{N};+,\cdot,\uparrow,1\rangle$ (up to a simple term equivalence), showing that the variety of $\langle \mathbb{N};+,\cdot,\uparrow,1\rangle$ 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 $\langle \mathbb{N};+,\cdot,\uparrow\rangle$.

发表机构

• Umm Al-Qura University（乌姆古拉大学）
• La Trobe University（乐卓博大学）
• University of Denver（丹佛大学）

机构由 AI 辅助整理，请以论文原文为准。