C-minimal theory
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In
model theory In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the s ...
, a branch of
mathematical logic Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...
, a C-minimal theory is a theory that is "minimal" with respect to a
ternary relation In mathematics, a ternary relation or triadic relation is a finitary relation in which the number of places in the relation is three. Ternary relations may also be referred to as 3-adic, 3-ary, 3-dimensional, or 3-place. Just as a binary relat ...
''C'' with certain properties. Algebraically closed fields with a (Krull) valuation are perhaps the most important example. This notion was defined in analogy to the o-minimal theories, which are "minimal" (in the same sense) with respect to a linear order.


Definition

A ''C''-relation is a ternary relation ''C''(''x'';''y,z'') that satisfies the following axioms. # \forall xyz\, C(x;y,z)\rightarrow C(x;z,y) # \forall xyz\, C(x;y,z)\rightarrow\neg C(y;x,z) # \forall xyzw\, C(x;y,z)\rightarrow (C(w;y,z)\vee C(x;w,z)) # \forall xy\, x\neq y \rightarrow \exists z\neq y\, C(x;y,z) A C-minimal structure is a
structure A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...
''M'', in a
signature A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...
containing the symbol ''C'', such that ''C'' satisfies the above axioms and every set of elements of ''M'' that is definable with parameters in ''M'' is a Boolean combination of instances of ''C'', i.e. of formulas of the form ''C''(''x'';''b,c''), where ''b'' and ''c'' are elements of ''M''. A theory is called C-minimal if all of its models are C-minimal. A structure is called strongly C-minimal if its theory is C-minimal. One can construct C-minimal structures which are not strongly C-minimal.


Example

For a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
''p'' and a ''p''-adic number ''a'', let , ''a'', ''p'' denote its ''p''-adic absolute value. Then the relation defined by C(a; b, c) \iff , b-c, _p < , a-c, _p is a ''C''-relation, and the theory of Q''p'' with addition and this relation is C-minimal. The theory of Q''p'' as a field, however, is not C-minimal.


References

* * {{Mathematical logic Model theory