  Generating vector representations (embeddings) of OWL ontologies is a growing
task due to its applications in predicting missing facts and knowledge-enhanced
learning in fields such as bioinformatics. The underlying semantics of OWL
ontologies is expressed using Description Logics (DLs). Initial approaches to
generate embeddings relied on constructing a graph out of ontologies,
neglecting the semantics of the logic therein. Recent semantic-preserving
embedding methods often target lightweight DL languages like
$\mathcal{EL}^{++}$, ignoring more expressive information in ontologies.
Although some approaches aim to embed more descriptive DLs like
$\mathcal{ALC}$, those methods require the existence of individuals, while many
real-world ontologies are devoid of them. We propose an ontology embedding
method for the $\mathcal{ALC}$ DL language that considers the lattice structure
of concept descriptions. We use connections between DL and Category Theory to
materialize the lattice structure and embed it using an order-preserving
embedding method. We show that our method outperforms state-of-the-art methods
in several knowledge base completion tasks. We make our code and data available
at https://github.com/bio-ontology-research-group/catE.
