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A Hausdorff topological group is called minimal if it does not admit a strictly coarser Hausdorff group topology. This paper mostly deals with the topological group $H_+(X)$ of order-preserving homeomorphisms of a compact linearly ordered connected space $X$. We provide a sufficient condition on $X$ under which the topological group $H_+(X)$ is minimal. This condition is satisfied, for example, by: the unit interval, the ordered square, the extended long line and the circle (endowed with its cyclic order). In fact, these groups are even $a$-minimal, meaning, in this setting, that the compact-open topology on $G$ is the smallest Hausdorff group topology on $G$. One of the key ideas is to verify that for such $X$ the Zariski and the Markov topologies on the group $H_+(X)$ coincide with the compact-open topology. The technique in this article is mainly based on a work of Gartside and Glyn.
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Classes SSGP(n)(n < omega) of topological groups are defined, and the class-theoretic inclusions SSGP(n) subseteq SSGP(n+1) subseteq m.a.p. are established and shown proper. These classes are investigated with respect to the properties normally studied by topologists (products, quotients, passage to dense subgroups, and the like). In passing the authors establish the presence of the SSGP(1) or SSGP(2) property in many of the early examples in the literature of abelian m.a.p. groups.
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