A Course of Mathematics. For the Use of Academies as Well by Charles Hutton

By Charles Hutton

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Extra resources for A Course of Mathematics. For the Use of Academies as Well as Private Tuition. Volume I (of Two)

Example text

In. Turning now to nonrepresentable varieties, we first consider a simple generalization of the abelian equation which proves to be very useful in classifying i-groups. For each n E Z+ , let Although these varieties are easily defined, they can be used to move all the way up the lattice structure of L. 12 (Smith [1980]) N = VL n . POWELL In contrast to the last theorem, Smith [1980] has also shown that /If is (finitely) join irreducible. A very striking result gives a complete description of the varieties between each Sp and Lp where p is prime.

6 The following varieties of i-groups fail the amalgamation property. (i) (Pierce) C (ii) (Pierce) All varieties U such that A2 ~ U. (iii) (Powell and Tsinakis) An uncountable chain Ua of varieties having intersection R. Also, all varieties V satisfying n nV =A and V c Ua (some 0:). These include each Sp and each Ln. (iv) (Powell and Tsinakis) The nilpotent varieties Nn(n > 1). "'12, n]. (vi) (Glass, Saracino, and Wood) The variety W of weakly abelian i-groups. For varieties of lattices, Day and Jezek [1984] have shown that any proper variety containing the variety of distributive lattices (which happens to be the smallest nontrivial lattice variety) must fail the amalgamation property.

In this case Glass [1984] and [1987] has shown that they can be represented in a nice way in terms of groups of permutations of totally ordered sets. We will outline here the construction of the A-free product since it has produced the most results. The generalization to other representable varieties is left to outside reading. ) Let G 1 , G 2 E A. POWELL A = {( (G I ill G2) / K, T) IK is a subgroup of G1 ill G2 and T is the positive cone of a total order extending the orders of G I and G 2 }. 2 (Powell and Tsinakis [1983]) Let G I , G2 E A and let A and 7jJ be as defined above.

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