MathStructures
http://math.chapman.edu/~jipsen/structures/
2019-04-20T05:33:19-07:00MathStructures
http://math.chapman.edu/~jipsen/structures/
http://math.chapman.edu/~jipsen/structures/lib/images/favicon.icotext/html2019-03-28T16:22:05-07:00Peter Jipsensemirings_with_zero
http://math.chapman.edu/~jipsen/structures/doku.php/semirings_with_zero?rev=1553815325&do=diff
Semirings with zero
Abbreviation: SRng$_0$
Definition
A <b><i>semiring with zero</i></b> is a structure of type such that
is a commutative monoid
is a semigroup
is a zero for : ,
distributes over : ,
Morphisms
Let and be semirings with zero. A morphism from
to is a function that is a homomorphism:text/html2019-03-28T16:15:01-07:00Peter Jipsensemirings_with_identity
http://math.chapman.edu/~jipsen/structures/doku.php/semirings_with_identity?rev=1553814901&do=diff
Semirings with identity
Abbreviation: SRng$_1$
Definition
A <b><i>semiring with identity</i></b> is a structure of type such that
is a commutative semigroup
is a monoid
distributes over : ,
Morphisms
Let and be semirings with zero. A morphism from
to is a function that is a homomorphism:text/html2019-03-28T15:01:59-07:00Peter Jipsenneofields
http://math.chapman.edu/~jipsen/structures/doku.php/neofields?rev=1553810519&do=diff
Neofileds
Abbreviation: Nfld
Definition
A <b><i>neofield</i></b> is a structure of type such that
is a loop
is a group
distributes over : and
Morphisms
Let and be neofields. A morphism from
to is a function that is a homomorphism:text/html2019-03-28T15:01:04-07:00Peter Jipsenleft_neofield
http://math.chapman.edu/~jipsen/structures/doku.php/left_neofield?rev=1553810464&do=diff
Left neofileds
Abbreviation: LNfld
Definition
A <b><i>left neofield</i></b> is a structure of type such that
is a loop
is a group
left-distributes over :
Morphisms
Let and be left neofields. A morphism from
to is a function that is a homomorphism:text/html2019-03-28T14:53:34-07:00Peter Jipsendivision_rings
http://math.chapman.edu/~jipsen/structures/doku.php/division_rings?rev=1553810014&do=diff
Division rings
Abbreviation: DRng
Definition
A <b><i>division ring</i></b> (also called <b><i>skew field</i></b>) is a ring with identity such that
is non-trivial:
every non-zero element has a multiplicative inverse:
Remark:
The inverse of is unique, and is usually denoted by .<b><i>A theorem on finite algebras</i></b><b>6</i></b>text/html2019-03-28T14:50:15-07:00Peter Jipsenindex.html
http://math.chapman.edu/~jipsen/structures/doku.php/index.html?rev=1553809815&do=diff
Mathematical Structures
The webpages collected here list information about classes of
mathematical structures. The aim is to have a central place to check
what properties are known about these structures.
These pages are currently still under construction. Knowledgeable readers are encouraged to add or correct information.
To enable the edit button on each page, use the Login link (above) to log in or create an account.text/html2019-03-14T23:06:50-07:00Peter Jipsensemifields
http://math.chapman.edu/~jipsen/structures/doku.php/semifields?rev=1552630010&do=diff
Semifields
Abbreviation: Sfld
Definition
A <b><i>semifield</i></b> is a semiring with identity such that
is a group, where if has an absorbtive , and otherwise.
Morphisms
Let and be semifields. A morphism from
to is a function that is a homomorphism:text/html2019-03-14T16:46:40-07:00Peter Jipsensemirings
http://math.chapman.edu/~jipsen/structures/doku.php/semirings?rev=1552607200&do=diff
Semirings
Abbreviation: SRng
Definition
A <b><i>semiring</i></b> is a structure of type such that
is a semigroup
is a commutative semigroup
distributes over : ,
Morphisms
Let and be semirings. A morphism from
to is a function that is a homomorphism:text/html2019-02-24T14:15:54-07:00Peter Jipsenresiduated_partially_ordered_monoids
http://math.chapman.edu/~jipsen/structures/doku.php/residuated_partially_ordered_monoids?rev=1551046554&do=diff
Residuated partially ordered monoids
Abbreviation: RpoMon
Definition
A <b><i>residuated partially ordered monoid</i></b> (or <b><i>rpo-monoid</i></b>) is a structure such that
is a partially ordered set,
is a monoid and
is the left residual of : <b><i>Title</i></b><b>1</i></b>text/html2019-02-24T11:39:49-07:00Peter Jipsenpocrims
http://math.chapman.edu/~jipsen/structures/doku.php/pocrims?rev=1551037189&do=diff
Pocrims
Abbreviation: Pocrim
Definition
A <b><i>pocrim</i></b> (short for <b><i>partially ordered commutative residuated integral monoid</i></b>) is a structure of type such that
(1):
(2):
(3):
(4):
(5):
Morphisms
Let and be pocrims. A morphism from to is a function that is a homomorphism:
,
,
.<b><i>pocrim</i></b><b><i>Dually residuated commutative monoids with identity element as least element do not form an equational class</i></b><b>29</i></b>