6 R. S. PIERCE a zero z, then necessarily ?(z) is the zero of 5 . But i f P i s not onto, ?(z) iray n 0 t ^ the z e r o °^ ^* I n what follows, we wil l always require of a semi-lattice homomorphism that i t map the zero element of i t s domain (i f such exists) onto zero. This convention can be considered a part of our definition of a semi-lattice homomorphism. The following property i s often useful for showing that a homomorphism of a disjunctive semi-lattice i s one-to-one. LEMM A 2.5 . Let S and 5 be send-lattices with zeros z and "z respectively. Assume that S i s disjunctive. Suppose V :S -* 5 i s a homomorphism with kernel {z\ (m 9~ ( {'z ) ) . Then «* i s one-to-one. PROOF. Since S i s disjunctive, i f a ^ b, there exists c 6 S such that z / c a, c A b » z. Then "z / ?(c) p(a) and f(c) A 4(b) - z. Hence, ?(a) ^ f ( b ) . 3. AN IMBEDDING THEOREM. DEFINITION 3 . 1 . Let P be a partially ordered set with zero element z. A subset S C P i s said t o be dense in P i f p / z in P implies that there i s an element s z S such that z / s p. PROPOSITION 3#2. Let D be a disjunctive semi-lattice and S a dense sub-semi-lattice of D. Then S is disjunctive. This i s easily seen from definitions 3»1 and 2.3 . Since any Boolean algebra is clearly disjunctive, i t follows from 3«2 that a dense sub-semi- lattic e of a BJl. is disjunctive. Somewhat more remarkable i s the converse. THEORE M 3 . 3 . Let D be a disjunctive seni-lattice . Then there i s an isomorphism p of D onto a dense sub-semi-lattice of a complete Boolean algebra. This theorem is a special case of more general results in several papers (see for example [3])» A simple proof of i t can be based on a well known theorem of Stone and Glivenko (see [7] or [Ik]) as follows. Let & be the set of a l l ideals of D. For A e £ , define A* « {b e D|a A b - z for al l a e A } Then the following facts can be
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