# Finite-to-finite universal quasivarieties are Q-universal

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## Abstract.

Let **K** be a quasivariety of algebraic systems of finite type. **K** is said to be *universal* if the category **G** of all directed graphs is isomorphic to a full subcategory of **K**. If an embedding of **G** may be effected by a functor **F**:**G** \( \longrightarrow \) **K** which assigns a finite algebraic system to each finite graph, then **K** is said to be *finite-to-finite* universal. **K** is said to be *Q-universal* if, for any quasivariety **M** of finite type, *L*(**M**) is a homomorphic image of a sublattice of *L*(**K**), where *L*(**M**) and *L*(**K**) are the lattices of quasivarieties contained in **M** and **K**, respectively.¶We establish a connection between these two, apparently unrelated, notions by showing that if **K** is finite-to-finite universal, then **K** is *Q*-universal. Using this connection a number of quasivarieties are shown to be *Q*-universal.

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