Pronunciation: [sˈɛmɪfˌiːld] (IPA) 
The word "semifield" is spelled with three syllables: /ˈsɛmɪfiːld/. The first syllable is pronounced as "sem" with a short "e" sound, the second syllable is pronounced as "i" as in "sit" and "fi" with a long "e" sound, and the final syllable is pronounced "ld" with a silent "e". This word does not have a common meaning in English, but it can be used to refer to a mathematical concept that relates to a subspace of a vector space.
A semifield is a mathematical structure that combines properties of both fields and nearrings. It is a set equipped with two binary operations, usually denoted as addition and multiplication, that satisfies a specific set of axioms. However, unlike a field, it does not necessarily contain inverses for every element under multiplication.
In more detail, a semifield is a nonempty set S together with two binary operations, generally denoted as + and ⋅, which satisfy the following conditions:
1. (S, +) forms a commutative group. This means that addition is associative, has a neutral element (often denoted as 0), and every element has an additive inverse.
2. (S\{0}, ⋅) forms a commutative monoid. This means that multiplication is associative, commutative, and has a neutral element (often denoted as 1).
3. The distributive property holds, that is, for all a, b, and c in S, a ⋅ (b + c) = (a ⋅ b) + (a ⋅ c).
4. For every nonzero element a in S, there exists a unique nonzero element b in S such that a ⋅ b = 1.
The lack of multiplicative inverses for all elements distinguishes semifields from fields. However, they still exhibit many properties of fields, making them useful in various mathematical structures and applications, particularly in coding theory and projective geometry.
The word "semifield" is derived from the combination of two words: "semi-" and "field".
The prefix "semi-" comes from the Latin word "semi", meaning "half", "part", or "incompletely". It is widely used in English to denote something that is partial, incomplete, or halfway.
The term "field" in mathematics refers to a set of elements on which two operations, usually addition and multiplication, are defined and follow specific rules. Fields have various applications in algebra, number theory, and other areas of mathematics.
Therefore, when merged, "semifield" denotes a mathematical structure or algebraic system that partially meets the properties and rules of a field. It represents a structure that exhibits some, but not all, of the properties of a field.
