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.
Several other lines of research touch tangentially on the ideas presented in this paper. The notion of dominance appears in @cite_14 @cite_10 as an integral part of the definition of . In @cite_14 @cite_10 , the condition is broader in the sense that it allows passing to submatrices and or making a global sign change. Also, the condition in @cite_14 @cite_10 is applied to extended exchange matrices, not just to exchange matrices. Some surface-cutting constructions similar to resection have appeared in @cite_19 @cite_10 , but resection is different because it aims to do something different, namely to construct dominance relations without restricting to submatrices and without freezing'' variables.
@cite_19 @cite_14 @cite_10
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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.