Re: Unicode Characters to display SKOS relations

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Useful sources:

Croft, William and D. Alan Cruse (2004). *Cognitive linguistics*. Cambridge
University Press.

Cruse, D. Alan. (1986). *Lexical semantics*. Cambridge University Press.

Riesthuis, Gerhard J. A. et al. (2008). *Guidelines for Multilingual
Thesauri*. IFLA Professional Reports,
No. 115. The Hague, NL: International Federation of Library Associations and
Institutions. URL:
http://www.ifla.org/VII/s29/pubs/Profrep115.pdf

Soergel, Dagobert (1974). Indexing languages and thesauri: construction and
maintenance. Information sciences series. Los Angeles: Melville Pub. Co.
ISBN: 0471810479.

In fact, read as much Soergel as you can find :-)

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I'm not sure if the semantics of SKOS are quite right for the mathematical
symbols  you're using

Let the type Δ denote the domain of discourse to which labels might be
attached.
Let the type Σ denote the set of all possible label strings .
Let the type CONCEPT denote a two-tuple, (Σ × ℙ(Δ))   containing a label and
a set of elements of Δ .
Let the type CONCEPT-SCHEME denote a  2-tuple (Σ  ×  ℙ(CONCEPT)).

Let c and k denote two arbitrary CONCEPTs
Let C and K denote two arbitrary CONCEPT-SCHEMEs

Let label(k) refer to the first element of CONCEPT k.
Let documents(k) refer to the second element of CONCEPT k.
Let label(C) refer to the first element of a CONCEPT-SCHEME C.
Let concepts(C) refer to the second element of a CONCEPT-SCHEME C.

Let the 2-tuple (c,C) denote a fully qualified concept (FQC) consisting of
of a concept and a concept scheme,  where c ∈ concepts(C).

BT, NT, and EQ for a single CONCEPT scheme.

Within a single CONCEPT-SCHEME C, such that c ∈ C ⋀ k ∈ C

1: ( BT)    c < k    iff documents(c) ⊂ documents(k)
2: (NT)    c > k     iff documents(c) ⊃ documents(k)
3: (SY)    c  ≍ k    iff documents(c) ≡ documents(k)

Unique Concept Scheme Name Assumption
4: ∀ C ∈ CONCEPT-SCHEME. ∀ K ∈ CONCEPT-SCHEME. label(C) ≡ label(D)  → C ≡ D

Within Scheme Unique Preferred Name Assumption
5: ∀ C ∈ CONCEPT-SCHEME. ∀ c ∈ concepts(C).  ∀ d ∈ concepts(C).  label(c) =
label(d) → c ≡ d

Identity
6: C = K    iff   label(c) ≡ label(k) ⋀  concepts(c) ≡ concepts(k)
7: c = k     iff   label(c) ≡ label(k) ⋀  documents(c) ≡ documents(k) ^ ∀ C
∈ CONCEPT-SCHEME. c ∈ concepts(C) iff k ∈ concepts(C)


Mapping Relations
   Note that mapping relations are only defined between concepts in
different concept schemes.

Exact match, (c,C) ≍ (k,K)
8: For an exact match, (c,C) ≍ (k,K)
    (c,C) ≍ (k,K) iff C ≢ K ⋀  documents(c) ≡ documents(k)

Broad Match:  (c,C)  ⪷ (k,K)

9: (c,C) ⪷ (k,K)  iff
              ¬ (c,C) ≍ (k,K) ⋀
               c < k  â‹€
              ∄d ∈ concepts(K). (c < d ⋀ d < k  ⋀ d ≠ k)

Narrower Match: (c,C) ⪸ (k,K)

10: (c,C) ⪸ (k,K)  iff
              ¬ (c,C) ≍ (k,K) ⋀
               c > k  â‹€
              ∄d ∈ concepts(K). (c > d ⋀ d > k  ⋀ d ≠ k)


Close Match:  (c,C) ≈ (k,K)

The semantics of close match are under determined:  as a bare minimum, we
must define a similarity function  f ∈ (CONCEPT × CONCEPT → [0,1]), together
with a threshold t below which two concepts are not considered to be a
match.

11:    (c,C) ≈ (k,K) iff  ¬ (c,C) ≍ (k,K) ⋀
                              f(c,k) ≥ t   ⋀
                              ∄d ∈ concepts(K). f(c,d) > f(c,k)