In this paper we give new criterions for left distributive posets to have neatest representations. We also illustrate a construction that would embed left distributive posets into representable semilattices. MSC: 06A10, 06A12.Any poset (P, 5) has at least two representations [l] as a poset (Q, S). One of these two representations has the property that the image of the supremum of any (finite or infinite) subset A of P is the union of the images of the elements of A, while the other representation has the property that the image of the infimum of any (finite or infinite) subset B of P is the intersection of the images of the elements of B. In general, we cannot have a representation which has both of the above mentioned properties. However, Smm (71 proved that i f ( P , 5) is a complemented distributive lattice, i.e., a Boolean Algebra, then it has a representation as a set (Q, E). BIRKHOFF (21 has proved a Stone type representation theorem where the lattice (P, 5) is only distributive. Along these lines, SCHEIN [6] proved that a poset in which the iniimum of every two elements exists and which has a special type of distributive property has a representation as a set (Q, E). There are other types of representations given in [3], [4], and[5]. In Section 1 of this paper, we give some other necessary and suficient conditions for posets with this type of distributivity to be representable. In Section 2, we generalize SCHEIN'S result and finally, in Section 3, we illustrate a construction that would embed existing posets into SCHEIN'S type of semilattices by adjoining infimum. With this procedure one could then show the representability of more general posets. 1. Neatest representations of posets D e f i n i t i o n 1. An order isomorphism f from a poset (P, 5) onto a poset (a S ) is called a neatest representation for P iff for every subset {ai: i E S) of P and for every u, b E P (i) /(Aic ai) = nie J ( a i ) , whenever A;. ui exists, (ii) f(o v b) = f ( a ) u f ( b ) , whenever a v b exists D e f i n i t i o n 2 . Let (P, s) be a poset. (P, s) is lefr meet distributive (1.m.d.) iff for every 19 ZUchr. f. math. Logik

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In this paper, we use the Lagrange neighbour and our equivalence theorem for primitive ideals to obtain lower bounds which are sharper than those given in the literature for class numbers of real quadratic fields in general, but applied to greatest advantage when d is of ERD type.

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