Semilattices and a ternary operation in modular lattices
S. A. Kiss · Bulletin of the American Mathematical Society · 1948
Before discussing the subject matter proper it is necessary to introduce the following: 1 LEMMA 1.The inequalityis identically satisfied in any lattice.and from these two inequalities followsFor purposes of facility of expression the concept of semilattice is here introduced following Klein-Barmen [l]: 2 DEFINITION 1.A semilattice L 8 is a partially ordered system in which a relation xay is defined which satisfies SI: For all x, xax, S2: If xay and yax, then x=*y, S3 : If xay and ycz, then xaz t and in which any two elements x and y have a greatest lower bound or meet xmy.It then follows that xmy or any binary operation xoy which is closed, idempotent, commutative and associative defines, by means of the convention that xay means xmy~x or xoy -x, a semilattice L 8 in which xmy or xoy is the greatest lower bound of x and y.LEMMA 2. The ternary operation (2) [*, /, y] = (xr\(t\Jy)) \J (tny) -(* U (*H y)) H (*U y)