Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-25T17:59:34.983Z Has data issue: false hasContentIssue false
  • English
  • Français

Axioms for Semi-Lattices

Published online by Cambridge University Press:  20 November 2018

D. H. Potts
D. H. Potts*
Affiliation:
University of California, Berkeley
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A semi-lattice (Birkhoff, Lattice Theory, p. 18, Ex. 1) is an algebra <A,.> with a single binary operation satisfying: (1) x = xx, (2) xy = yx, and (3) (xy)z = x(yz). In this note we show that the three identities may be reduced to two but cannot be reduced to one.

It is easy to see that (2), (3) imply (4) (uv)((wx)(yz)) = ((vu)(xw))(zy). Setting w = y = u and x = z = v in (4) and using (1) we get uv = vu. Setting v = u, x = w, and z =y in (4) and using (1) we get u(wy) = (uw)y. And so (1) and (4) imply (2) and (3).

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 8 , Issue 4 , June 1965 , pp. 519
Copyright
Copyright © Canadian Mathematical Society 1965