Introduction. This paper is concerned with the problem of imbedding a Boolean algebra B into an ^-complete Boolean algebra 5* in such a way that certain homomorphisms of B can be extended to B*. We investigate two such imbeddings which arose naturally from the consideration of the work of Rieger and Sikorski in [5] and [7]. In [5] Rieger proved the existence of a certain class of free Boolean algebras and investigated their representability by α-fields of sets. Rieger's theorem on the existence of "the free α-complete Boolean algebra on m generators" is equivalent to the following statement: Every free Boolean algebra B can be imbedded in an ^-complete Boolean algebra J5* such that every homomorphism of B into an incomplete Boolean algebra C can be extended to an ^-homomorphism of B* into C. The question now arises: Does this result hold for an arbitrary Boolean algebra B which is not necessarily free? If such an imbedding exists, we call B* the free a-extensίon of B.In [7], Sikorski gave a characterization of all the σ-regular extensions of a Boolean algebra B. To obtain this characterization, he first proved that B can be imbedded as a σ-regular subalgebra of â -complete Boolean algebra B * such that every α-homomorphism of B into a σ-complete Boolean algebra C can be extended to a ^-homomorphism of B* into C. We call J5* the free σ-regular extension of B.In § 2 of this paper we prove that the free α-extension B Λ of B exists uniquely for every Boolean algebra B and every infinite cardinal number a. In § 3 we investigate the representability of B Λ by an α-field of sets. We first prove that B a is isomorphic to an α-field of sets if and only if it is α-representable. A corollary to this result is that the free σ-extension B σ of an arbitrary Boolean algebra B is isomorphic to a σ-field of sets. The problem of characterizing those Boolean algebras B for which B ω is α-representable for a ^ 2Ko is also discussed. In §4 we investigate the α:-regular extensions of Boolean algebras for an arbitrary cardinal number α. Sikorski's results on the σ-regular extensions depend on the Loomis-Sikorski theorem which does not hold for uncountable cardinal numbers. We use our results on the free α-extension B^ of B to prove the existence of the free α-regular extension and to give a characterization of the α-regular

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It is known that every Boolean algebra B satisfies the infinite distributive laws while an arbitrary distributive lattice may fail to satisfy either or both of these laws. In this note we show that both (DO and (Dz) hold for a Post L-algebra P= (B, L) with a finite lattice of constants L. Post L-algebras were introduced by T. P. Speed in [2] and further investigated by the author in [3] and [4]. It is shown in [2] that if L is a bounded distributive lattice, then every Post L-algebra P is isomorphic to the coproduct of a Boolean algebra B and L, where the coproduct is taken in the category of bounded distributive lattices and lattice homomorphisms preserving 0 and 1. P will be denoted by P= (B, L).All lattices considered in this note will be distributive lattices with 0 and 1, and all lattice homomorphisms will preserve 0 and 1. We shall use the terminology and notation of [1]. In particular, if L' is a sublattice of L and SC=L ", then the least L' upper bound of S in L' and L will be denoted (whenever they exist) by Z x and xES L Z x respectively, similar notation will be used for the greatest lower bounds of S xES in L' and L. We recall the definition of coproduct.DEFINrrIoN. Let L1, L2 and L be distributive lattices and let ix: Lx-~L and i,,: Ls-~ L be lattice monomorphisms. ~Ihe pair (L, {iI, is}) will be called the coproduet (=free product) of L1 and Ls ff for every distributive lattice D and every pair of lattice homomorphisms hi:/--1 ~ D and hs: Ls-~D, there is a unique lattice homomorphism h: L~ D such that hi~ = h~ and his = hs.We shall denote the coproduct of L~ and Ls by/,1. Ls and to simplify the notation we shall identify Lx and L2 with their isomorphic images ix(/-a) and is (La)

There are a number of classes of distributive lattices whose members can be characterised as the coproduct A In this note we investigate the a-representability of the coproduct A * L of two distributive lattices. In Section 2 we show (Theorem 2.3) that if L is finite, then A * L is a-complete if and only if A is a-complete, and (Theorem 2.6) if L is arbitrary and B is a Boolean algebra, then B * L is a-complete if and only if both B and L are a-complete and at least one of them is finite. The a-representability of A * L, where L is finite, is discussed in Section 3 where we show (Theorem 3.2) that A * L is an a-homomorphic image of an a-ring of sets if and only if A has the same property, and (Theorem 3.5) A * L is isomorphic to an a-ring of sets modulo an a-ideal if and only if A has the same property. The specialisation of these results to Post algebras and their generalisations yields the known, as well as some new, results concerning the a-representability of these algebras, (see Corollary 3.4 and Remark 3.6.) NotationAll lattices considered below will be distributive lattices with 0 and 1 and all lattice homomorphisms will preserve 0 and 1. The least upper bound and greatest lower bound of two elements x and y will be denoted by x + y and xy respectively. If L' is a sublattice of L and SsL', then the least upper bounds of S in L and L will be denoted