Geodesic
Axioms for Semi-Lattices
Canadian mathematical bulletin, Tome 8 (1965) no. 4, p. 519

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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). 
Potts, D. H. Axioms for Semi-Lattices. Canadian mathematical bulletin, Tome 8 (1965) no. 4, p. 519. doi: 10.4153/CMB-1965-039-9
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     doi = {10.4153/CMB-1965-039-9},
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