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RINGS WITH COMMON DIVISION, COMMON MEADOWS AND THEIR CONDITIONAL EQUATIONAL THEORIES

Part of: Varieties Other classes of algebra Connections with logic Algebraic structures

Published online by Cambridge University Press:  23 December 2024

JAN A. BERGSTRA Open the ORCID record for JAN A. BERGSTRA [Opens in a new window]  and
JOHN V. TUCKER Open the ORCID record for JOHN V. TUCKER [Opens in a new window]
JAN A. BERGSTRA
Affiliation:
INFORMATICS INSTITUTE, UNIVERSITY OF AMSTERDAM SCIENCE PARK 900, 1098 XH, AMSTERDAM THE NETHERLANDS E-mail: j.a.bergstra@uva.nl
JOHN V. TUCKER*
Affiliation:
DEPARTMENT OF COMPUTER SCIENCE SWANSEA UNIVERSITY, BAY CAMPUS FABIAN WAY, SWANSEA, SA1 8EN UK
*
E-mail: j.v.tucker@swansea.ac.uk
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Abstract

We examine the consequences of having a total division operation $\frac {x}{y}$ on commutative rings. We consider two forms of binary division, one derived from a unary inverse, the other defined directly as a general operation; each are made total by setting $1/0$ equal to an error value $\bot $, which is added to the ring. Such totalised divisions we call common divisions. In a field the two forms are equivalent and we have a finite equational axiomatisation E that is complete for the equational theory of fields equipped with common division, which are called common meadows. These equational axioms E turn out to be true of commutative rings with common division but only when defined via inverses. We explore these axioms E and their role in seeking a completeness theorem for the conditional equational theory of common meadows. We prove they are complete for the conditional equational theory of commutative rings with inverse based common division. By adding a new proof rule, we can prove a completeness theorem for the conditional equational theory of common meadows. Although, the equational axioms E fail with common division defined directly, we observe that the direct division does satisfy the equations in E under a new congruence for partial terms called eager equality.

Keywords

commutative ringsdivision operatorsmeadowscommon divisioncommon meadowsequationsconditional equationsvarietiesquasivarieties

MSC classification

Primary: 08B05: Equational logic, Mal′cev (Mal′tsev) conditions 08C15: Quasivarieties 12L12: Model theory
Secondary: 08A55: Partial algebras

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Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 31
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

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