Ontology operations, e.g., aligning and merging, were studied and implemented extensively in different settings, such as, categorical operations, relation algebras, typed graph grammars, with different concerns. However, aligning and merging operations in the settings share some generic properties, e.g., idempotence, commutativity, associativity, and representativity, labeled by (I), (C), (A), and (R), respectively, which are defined on an ontology merging system $(\mathfrak{O},\sim,\merge)$, where $\mathfrak{O}$ is a set of the ontologies concerned, $\sim$ is a binary relation on $\mathfrak{O}$ modeling ontology aligning and $\merge$ is a partial binary operation on $\mathfrak{O}$ modeling ontology merging. Given an ontology repository, a finite set $\mathbb{O}\subseteq \mathfrak{O}$, its merging closure $\widehat{\mathbb{O}}$ is the smallest set of ontologies, which contains the repository and is closed with respect to merging. If (I), (C), (A), and (R) are satisfied, then both $\mathfrak{O}$ and $\widehat{\mathbb{O}}$ are partially ordered naturally by merging, $\widehat{\mathbb{O}}$ is finite and can be computed efficiently, including sorting, selecting, and querying some specific elements, e.g., maximal ontologies and minimal ontologies. We also show that the ontology merging system, given by ontology $V$-alignment pairs and pushouts, satisfies the properties: (I), (C), (A), and (R) so that the merging system is partially ordered and the merging closure of a given repository with respect to pushouts can be computed efficiently.

翻译： Ontolog 操作, 例如对齐和合并, 在不同场合广泛研究和实施, 比如, 绝对操作, 关系代数, 图形语法, 不同关注 。 但是, 设置中 的对齐和合并操作具有一些通用属性, 例如 : 一元能力、 通性、 关联性和代表性, 分别由 (I) 、 (C) 、 (A) 和 (R) 分别标注, 定义于一个本体合并系统 $ (mathfrak{ O}, comm, commerge) $, 其中 $\ mayfrak{ O} 。 但是, $\ sim 是一个双元关系, 以( mathfrak} 标注 ) 和 美元 立體數值為 。 在 Oralfrequestal 上, (Oral_ blickral_ ) 也可以使用 立定義 。 (Oral_ b) 和 立體 立體 立體 立體 。