Reviewer has chosen to be Anonymous

Overall Impression: Good

Content:
Technical Quality of the paper: Good
Originality of the paper: Yes
Adequacy of the bibliography: Yes

Presentation:
Adequacy of the abstract: Yes
Introduction: background and motivation: Good
Organization of the paper: Satisfactory
Level of English: Satisfactory
Overall presentation: Excellent

Detailed Comments:

This paper presents a new geometric embedding method based on order embedding for knowledge bases of Description Logic ALC. It includes two steps: (1) transforming the axioms/ontology to a DAG composed of subsubsumption relationships of sub-expressions (which include the named concepts and the other expressions like the concept conjunction and disjunction); (2) using an order embedding algorithm to embed the nodes in the DAG. The paper is well written and easy to follow with good technical quality. The idea is novel, and the codes look to be well maintained.

In Line 29, Page 5, "The lattice construction is not compete since we consider a subset $\hat{\bm{C}}$ from the infinite set $\bm{C}$ of possible concept descriptions". Could the authors give more justification? What concept descriptions could be ignored? What problem will this lead to?

In Theorem 2, the authors proves that the embedding (mapping function $f_{\theta}$) is a lattice preserving function of (\hat{\bm{C}}, \subseteq). But how about preservation of the semantics of the original ontology $\mathcal{O}$? The author can give some justification on whether the embedding is sound enough to keep the semantics of the original ontology with some specific settings in learning. If not, why, and what semantics cannot be preserved?

In the second step, beside order embedding algorithms, will authors consider some other hierarchy embedding methods such as poincare ball-based hyperbolic embedding methods and box/cone-based geometric embedding methods? Please justify (and implement and evaluate if possible) its potential extensions.

ELEmbedding and Box^2EL are for ontologies of DL EL++. How are axioms beyond DL EL++ processed? Do GO and FoodOn include axioms beyond DL ALC? If yes, how are they processed? In Table 1, please give the DL languages (expressivities) of the ontologies used for experiments.

In Line 2, Page 6, "correspond" --> "corresponds".

Reviewer has chosen not to be Anonymous

Overall Impression: Average

Content:
Technical Quality of the paper: Average
Originality of the paper: Yes, but limited
Adequacy of the bibliography: Yes, but see detailed comments

Presentation:
Adequacy of the abstract: Yes
Introduction: background and motivation: Good
Organization of the paper: Satisfactory
Level of English: Satisfactory
Overall presentation: Good

Detailed Comments:

The paper is interesting and the results overall show the benefit of the approach. The paper heavily leverages the work by C. Le Duc (reference 27, in this paper), which, in part, limits the theoretical novelties of this paper. That said, the numerical results are convincing and the code has been made available.

I would recommend to split the long introduction by introducing a subsection towards the end where the authors describe the proposed approach and the benefits. That helps the readers to jump to the main part of in case they are familiar with the related work.

At some level, there is a bit of a question if one should consider the approach category theory based, given that in the end the authors are really using the algebraic structure of a lattice. Sure, a poset is a category, but at some levels the use of this very general terminology in this specific setting might be counter productive, i.e. "scare" possible readers away. I am not suggesting the authors to make changes, just making a general comment.

Theorem 1 is trivial and does not really warrant a proof and could be condensed to a "Follows by checking the properties of a poset."

Some minor suggestions:

Page 5 line 34, why do you put "saturation" in quotes? It has been used in other previous parts without quotation.
Page 5 line 36, 'inmediate' -> 'immediate'
Page 6 line 4, 'per morphism' or 'for each morphism'
Page 6 line 4 , 'Due to the large space complexity of Equation 5' -> 'Due to the large space complexity required when implementing Equation 5'.