We explore new interactions between finite model theory and classical streams of universal algebra and semigroup theory. A key result is an example of finite algebras whose variety is not finitely axiomatisable in first order logic, but where the class of finite members are finitely axiomatisable amongst finite algebras. These algebras present a negative solution to a first order formulation of the Eilenberg-Schützenberger problem, and witness the simultaneous failure of the Łos-Tarski Theorem, the SP-Preservation Theorem and Birkhoff's HSP-Preservation Theorem at the finite level. The examples also show that a pseudovariety without any finite pseudoequational basis may be finitely axiomatisable in first order logic amongst finite algebras. Other results include the undecidability of deciding first order definability of the pseudovariety of a finite algebra, and a mapping from any fixed finite template constraint satisfaction problem to a first order equivalent variety membership problem.

翻译：我们探索了有限模型论与泛代数及半群理论经典流派之间的新互动。一个关键结果是给出了有限代数的实例，其簇在一阶逻辑中无法有限公理化，但其中有限成员类在有限代数中可有限公理化。这些代数对Eilenberg-Schützenberger问题的一阶表述给出了否定解，并见证了Łos-Tarski定理、SP保持定理以及Birkhoff的HSP保持定理在有限层面同时失效。这些实例还表明，不存在任何有限伪等式基的伪簇可能在有限代数中可通过一阶逻辑有限公理化。其他结果包括判定有限代数伪簇的一阶可定义性的不可判定性，以及从任意固定有限模板约束满足问题到一阶等价簇成员判定问题的映射构造。