Forbidden Forests in the Prime Ideal Spectra of Lattice-Ordered Groups
by
R. N. Ball
University of Denver

Let Gamma(G) designate the poset of prime convex l-subgroups of an l-group G. We are interested in those l-groups for which Gamma(G) contains no copy of a given finite poset T. We show that all such l-groups G are characterized by the satisfaction in C_p(G), the lattice of principal convex l-subgroups of G, of a particular first-order sentence in the language of lattice theory. (If for a given l-group G there is no order embedding of a finite poset T into Gamma(G), we say that T is forbidden in Gamma(G). Of course, Gamma(G) is a forest (aka root system), and so the only posets which embed in it are also forests. This explains the title of the abstract.)

Theorem: For every finite poset T there is a first-order sentence theta in the language of lattice theory such that for every l-group G, theta holds in the lattice C_p(G) iff Gamma(G) contains no copy of T. Furthermore, if T is connected, i.e., if T is a tree, then theta has the form for all, exists varphi, where varphi is a conjunction of lattice equations.

Here are some examples. In what follows, G is an l-group with Gamma=Gamma(G), elements labeled a and c lie in G⁺, and [^a] designates the convex l-subgroup of G generated by a.

Proposition: Gamma has depth strictly less than n, i.e., Gamma contains no copy of an (n+1)-element chain, iff for all a_i there exist c_i such that
(1) [^a]₀ /\ [^c]₀=[^0],
(2) [^a]_k /\ [^c]_k <= [^a]_k-1 \/ [^c]_k-1, for all 0<k<n,
(3) [^a]_n <= [^a]_n-1 \/ [^c]_n-1.

Proposition: Gamma has width strictly less than n, i.e., Gamma contains no n-element antichain, iff for all a_i there is an index k such that [^a]_k <= \/ _{i =/= k}[^a]_i.

Proposition: Gamma has branching strictly less than n, i.e., Gamma contains no bounded n-element antichain, iff for all a_i there exist c_i such that
(1) a_i /\ c_i <= \/ _{i =/= k}a_i, for all 1 <= i <= n;
(2) a₀ <= \/ _{1 <= i <= n}( a_i \/ c_i).

Date received: December 26, 2002