W. Charles Holland

Top Varieties of Generalized MV-Algebras and Unital Lattice-Ordered Groups

In spite of the well-know fact that the system of ℓ-groups with strong unit (unital ℓ-groups) does not form a variety, there is a categorical connection between the category of unital ℓ-groups and the variety of generalized MV-algebras which enables us to naturally export equational machinery and terminology like “variety” from the latter category to the former. Using this categorical equivalence, we study varieties, or equationally defined classes, and top varieties, varieties above the normal valued variety, of both structures. We generalize Changs Completeness Theorem for generalized MV-algebras, and formulate some open questions for both structures.

Varieties of lattice-ordered groups in which prime powers commute

We show that the set of all varieties of lattice ordered groups contained in the metabelian variety defined by the lawxpyp=ypxp (wherep is a prime) is a well ordered tower. We give an explicit construction of the subdirectly irreducible members of each of these varieties and show that each variety is defined by a single equation.

Intrinsic metrics for lattice ordered groups

Swamy and Jakubik studied the metric ¦x y¦ on lattice ordered groups, and isometries which presere it. We show the only intrinsic metrics on lattice ordered groups are the multiplesn ¦x−y ¦ of theirs, and that the triangle inequality is satisfied by such a metric iff the group is abelian. We show that there are isometries for each of these metrics, but they are rare. We give a simpler proof via permutation groups of the following augmented version of a theorem of Jakubik. IfT is an isometry of the lattice ordered groupG with respect to the metric ¦x¥¦ andT(0)=0, thenG=A⊕B, B is abelian, andT(a+b)=a−b; conversely, any suchT is an isometry.

A very large class of small varieties of lattice-ordered groups 1

There are uncountably many varieties of lattice-ordered groups which cover the Abelian variety

Rigid homogeneous chains

Classifying (unordered) sets by the elementary (first order) properties of their automorphism groups was undertaken in (7), (9) and (11). For example, if Ω is a set whose automorphism group, S (Ω), satisfies then Ω has cardinality at most ℵ 0 and conversely (see (7)). We are interested in classifying homogeneous totally ordered sets (homogeneous chains, for short) by the elementary properties of their automorphism groups. (Note that we use ‘homogeneous’ here to mean that the automorphism group is transitive.) This study was begun in (4) and (5). For any set Ω, S (Ω) is primitive (i.e. has no congruences). However, the automorphism group of a homogeneous chain need not be o -primitive (i.e. it may have convex congruences). Fortunately, ‘ o -primitive’ is a property that can be captured by a first order sentence for automorphisms of homogeneous chains. Hence our general problem falls naturally into two parts. The first is to classify (first order) the homogeneous chains whose automorphism groups are o -primitive; the second is to determine how the o -primitive components are related for arbitrary homogeneous chains whose automorphism groups are elementarily equivalent.

Metabelian varieties ofl-groups which contain no non-abeliano-groups

The aim of this paper is to describe the lattice of all varieties of lattice ordered groups which are solvable class two (that is,metabelian) as groups and which have the property that they contain no non-abeliano-groups. In doing so we obtain a fairly explicit description of the structure of any finitely generated subdirectly irreducible member of such a variety and show that each such member generates a variety determined by a single identity.This continues the work of [HR] and [HMR]. In [S], Scrimger discovered an infinite set of varieties which cover the abelian variety, each of which is metabelian and contains only abeliano-groups. In [HR] the sub-directly irreducible members of the Scrimger varieties were characterized and from that a basis of identities was obtained. In [HMR] and independently in [G], those results were extended to obtain a description of the subvarieties of the varietyℒp of metabelianl-groups satisfying the identityxpyp=ypxp (p a prime).

Varieties of automorphism groups of orders

The group A(ST) of automorphisms of a totally ordered set fi must generate either the variety of all groups or the solvable variety of class n. In the former case, A(fi) contains a free group of rank 2N° ; in the latter case, A(ST) contains a free solvable group of class n — 1 and rank 2K°.

Outer automorphisms of ordered permutation groups

tyW. CHARLES HOLLAND(Received 18th December 1973)1. IntroductionA well-known fact is that every automorphism of the symmetric group on aset must be inner (whether the set is finite or infinite) unless the set has exactlysix elements (4, § 13). A long-standing conjecture concerns the analogue of thisfact for the group A(S) o alfl order-preserving permutations o af totally orderedset S. The group A(S) is lattice ordered (l-group) by defining, forf,ge A(S), f^g whenever xf ^ xg for all xeS. From the standpoint of/-groups, A(S) is of considerable interest because of the analogue of Cayleystheorem proved in (2), namely every /-group may be embedded in some A(S).Unlike the non-ordered symmetric groups, which must be highly transitive,A(S) is severely restricted if assumed transitive. Of special interest are thosecases when A(S) is doubly transitive (relative to the order), since the buildingblocks (primitive) of all transitive A(S) are either doubly transitive or uniquelytransitive. It is easily seen that each inner automorphism of A(S) must preservethe lattice ordering (that is, mus l-automorphism).t be an It is tempting toconjecture that every /-automorphism of A(S) must be inner. However, aneasy counterexample is at hand. If S consists of two copies of the ordered setof integers on, e entirely above the other, then A(S) is the direct product of twocopies of the /-group of integers, which has an outer /-automorphism exchangingthe two factors. The conjecture referred to above is that if A(S) is transitiveon S, then every /-automorphism of A(S) is inner. The conjecture has beenverified for many special classes of groups A(S) satisfying stronger hypothesesthan transitivity ((1), (5), (7), (8)). In this paper it is shown that the conjectureis false. In fact, there is an example of A(S) which is not only transitive butdoubly transitive, and which has an outer /-automorphism. Some slightlyweaker conjectures are investigated, and it is found that for a transitive A(S),every closed /-ideal is /-characteristic, and in most, but not all, cases, every/-ideal is /-characteristic.2. A counterexampleWe will construct a totally ordered set S which is doubly homogeneous (forevery x, y, z, w e S, if x < y and z < w, then there exists g e A (S) such that xg = zand yg = w) and such that A(S) has an outer /-automorphism. I am indebted

Covers of the Abelian Variety of Generalized MV-Algebras

Using the categorical equivalence of the class of generalized MV-algebras with the class of unital ℓ-groups, we describe all varieties of symmetric top abelian unital ℓ-groups that cover the variety 𝒜 uℓ of abelian unital ℓ-groups. Equivalently, we describe all cover varieties of the variety of MV-algebras, ℳ𝒱, within the variety of generalized MV-algebras admitting only one negation and each of whose maximal ideals is normal. In particular, there are continuum many representable varieties of generalized MV-algebras that cover ℳ𝒱.

Lattice-Ordered Permutation Groups

Although most of the elementary theorems about o-groups are as easy to prove in the general case as in the commutative case, there are no natural examples of non-commutative o-groups. Of course, it is easy to construct examples of non-commutative o-groups, but all of these are artificial and without much interest outside of the context of ordered groups. The situation is quite different for general lattice-ordered groups, however, because there is a class of non-commutative examples of great intrinsic interest, independent of the fact that they are lattice-ordered. These are the groups A(Ω) of order-preserving permutations of totally ordered sets Ω, endowed with the pointwise order. This means that for f,g ∈ A(Ω), we declare that f ≤ g iff for all α ∈ Ω, αf≤ αg. It is easily checked that this makes A(Ω) a lattice-ordered group in which α(f∧g) = αf∧αg. Only rarely is A(Ω) commutative.