Axioms for Semi-Lattices
- 1 June 1965
- journal article
- Published by Canadian Mathematical Society in Canadian Mathematical Bulletin
- Vol. 8 (4) , 519
- https://doi.org/10.4153/cmb-1965-039-9
Abstract
A semi-lattice (Birkhoff, Lattice Theory, p. 18, Ex. 1) is an algebra 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).Keywords
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