We give an algebraic characterisation of first-order logic with the neighbour relation, on finite words. For this, we consider languages of finite words over alphabets with an involution on them. The natural algebras for such languages are involution semigroups. To characterise the logic, we define a special kind of semidirect product of involution semigroups, called the locally hermitian product. The characterisation theorem for FO with neighbour states that a language is definable in the logic if and only if it is recognised by a locally hermitian product of an aperiodic commutative involution semigroup, and a locally trivial involution semigroup. We then define the notion of involution varieties of languages, namely classes of languages closed under Boolean operations, quotients, involution, and inverse images of involutory morphisms. An Eilenberg-type correspondence is established between involution varieties of languages and pseudovarieties of involution semigroups.

中文翻译：

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具有邻域的一阶逻辑的代数刻画

我们在有限词上给出了具有邻居关系的一阶逻辑的代数刻画。为此，我们考虑有限单词的语言而不是字母，并对其进行对合。这种语言的自然代数是对合半群。为了描述逻辑，我们定义了对合半群的一种特殊的半直接乘积，称为局部厄米积。具有邻域的FO的定理定理指出，一种语言在逻辑中是可定义的，并且仅当该语言被非周期性可交换对合半群和局部琐碎对合半群的局部Hermitian积识别时才可定义。然后，我们定义语言的对合变种的概念，即在布尔运算，商数，对合和对合态射影的逆像下封闭的语言类别。