The Representation of Lattices by Modules
نویسندگان
چکیده
That is, the class JSf(A) of lattices representable by A-modules is the "quasivariety" of lattices satisfying J(A), for commutative A. OUTLINE OF PROOF. For A commutative, let i : L > r ( M ; A ) be an embedding for some M. Without loss of generality, assume that L has a smallest element co, and i{œ) = 0. Motivated by the "abelian" lattice Tf{G ) of [2,4.2] with G = M, we consider "constraint systems" in variables ak (corresponding to coordinate positions in M ) and "auxiliary" variables bk (with existential quantifiers understood) for k in N = {1,2, 3 , . . .} . Consider r = (d1,d2,d3,d4) below. (d^ a1ex1, a2ex2, akea> for ^ 3 ( x 1 , x 2 e L ) . (d2) b 1 e x 3 , b2ex1, bkew for / c ^ 3 ( x 3 e L ) . (d3) axa2-bl= 0. (d4) a t V ? 2 = 0 (A0eA). A "solution" f.N -• M of r satisfies
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