bibkey: geiger1968closed
authors: "David Geiger"
year: 1968
title: "Closed systems of functions and predicates"
doi: 10.2140/pjm.1968.27.95
url: https://msp.org/pjm/1968/27-1/p08.xhtml
claim: "Closure of relations under primitive-positive definitions and preservation by operations form the classical relational Galois framework."
strata_touched:
  - D5/S3/Combinatorics/ConstraintRelations/PrimitivePositive
  - D5/S3/Combinatorics/ConstraintRelations/BooleanNullBoundary
license: citation-only
triage: anchor

Closed systems of functions and predicates

The preservation implication is proved directly by induction on finite atom lists in PrimitivePositive. The converse Galois theorem is not used.

Verified locator

• DOI: https://doi.org/10.2140/pjm.1968.27.95
• URL: https://msp.org/pjm/1968/27-1/p08.xhtml

The publisher PDF https://msp.org/pjm/1968/27-1/pjm-v27-n1-p08-p.pdf was retrieved (802,538 bytes; 16,054 extracted text characters). Pages 95-98 define commutation, variable identification and existential closure; the paragraph before Theorem 3 gives the preservation implication. Page 99 connects the framework to Post’s classification. The finite-domain converse is background; the preservation direction proved here also applies to arbitrary carriers.