An exchange matrix @math dominates an exchange matrix @math if the signs of corresponding entries weakly agree, with the entry of @math always having weakly greater absolute value. When @math dominates @math , interesting things happen in many cases (but not always): the identity map between the associated mutation-linear structures is often mutation-linear; the mutation fan for @math often refines the mutation fan for @math ; the scattering (diagram) fan for @math often refines the scattering fan for @math ; and there is often an injective homomorphism from the principal-coefficients cluster algebra for @math to the principal-coefficients cluster algebra for @math , preserving @math -vectors and sending the set of cluster variables for @math (or an analogous larger set) into the set of cluster variables for @math (or an analogous larger set). The scope of the description "often" is not the same in all four contexts and is not settled in any of them. In this paper, we prove theorems that provide examples of these dominance phenomena.
All of the references cited just above are concerned primarily with notions of morphisms of cluster algebras, but, to the author's knowledge, the homomorphisms proposed here do not fit into any category of cluster algebras already proposed. One category, proposed in @cite_19 , features . These require that initial cluster variables (including frozen'' variables) map to cluster variables or to integers. In our examples, the map restricts to a bijection on (non-frozen) initial cluster variables, but the frozen variables (which we call and denote by @math ) may map to other elements of the cluster algebra. In @cite_14 , it is shown that each rooted cluster morphism defines a seed homomorphism, or, loosely speaking, a dominance relationship in the broader sense allowing submatrices and or a sign change as above. Other notions of morphisms between cluster algebras include the of [Section 12] ca4 and [Section 3] universal and the of @cite_7 , both of which fix the exchange matrix @math .
@cite_19 @cite_14 @cite_7
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We introduce a category of cluster algebras with fixed initial seeds. This category has countable coproducts, which can be constructed combinatorially, but no products. We characterise isomorphisms and monomorphisms in this category and provide combinatorial methods for constructing special classes of monomorphisms and epimorphisms. In the case of cluster algebras from surfaces, we describe interactions between this category and the geometry of the surfaces. Our motivation is to build a systematic method in order to investigate the structure of cluster algebras of geometric type. The method is given through the notion of mixing-type sub-seeds, the theory of seed homomorphisms and the view-point of gluing of seeds. As an application, for (rooted) cluster algebras, we completely classify rooted cluster subalgebras and characterize rooted cluster quotient algebras in detail. Also, we build the relationship between the categorification of a rooted cluster algebra and that of its rooted cluster subalgebras. Note that cluster algebras of geometric type studied here are of the sign-skew-symmetric case. We introduce quasi-homomorphisms of cluster algebras, a flexible notion of a map between cluster algebras of the same type (but with different coefficients). The definition is given in terms of seed orbits, the smallest equivalence classes of seeds on which the mutation rules for non-normalized seeds are unambiguous. We present examples of quasi-homomorphisms involving familiar cluster algebras, such as cluster structures on Grassmannians, and those associated with marked surfaces with boundary. We explore the related notion of a quasi-automorphism, and compare the resulting group with other groups of symmetries of cluster structures. For cluster algebras from surfaces, we determine the subgroup of quasi-automorphisms inside the tagged mapping class group of the surface.