Year: 2024
Type: article
Abstract: In 1990, Johnstone gave a syntactic characterisation of the equational theories whose associated varieties are cartesian closed. Among such theories are all unary theories—whose models are sets equipped with an action by a monoid M—and all hyperaffine theories—whose models are sets with an action by a Boolean algebra B. We improve on Johnstone’s result by showing that an equational theory is cartesian closed just when its operations have a unique hyperaffine–unary decomposition. It follows that any non-degenerate cartesian closed variety is a variety of sets equipped with compatible actions by a monoid M and a Boolean algebra B; this is the classification theorem of the title.
Source: Algebra Universalis
Author: Richard Garner
Institution: Macquarie University
Language: English
Topic: Advanced Algebra and Logic
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
Funder: Macquarie University
OpenAlex ID: W4402519925