MathStructures
http://math.chapman.edu/~jipsen/structures/
2019-10-15T04:23:02-07:00MathStructures
http://math.chapman.edu/~jipsen/structures/
http://math.chapman.edu/~jipsen/structures/lib/images/favicon.icotext/html2019-10-13T18:50:41-07:00Peter Jipsenquantales
http://math.chapman.edu/~jipsen/structures/doku.php/quantales?rev=1571017841&do=diff
Quantales
Abbreviation: Quant
Definition
A <b><i>quantale</i></b> is a structure of type such that
is a complete semilattice with ,
is a semigroup, and
distributes over : and
Remark: In particular, distributes over the empty join, so .<b><i>Title</i></b><b>1</i></b>text/html2019-07-25T15:03:25-07:00Peter Jipsenindex.html
http://math.chapman.edu/~jipsen/structures/doku.php/index.html?rev=1564092205&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/html2019-07-20T10:48:42-07:00Peter Jipsendistributive_residuated_lattices
http://math.chapman.edu/~jipsen/structures/doku.php/distributive_residuated_lattices?rev=1563644922&do=diff
Distributive residuated lattices
Abbreviation: DRL
Definition
A <b><i>distributive residuated lattice</i></b> is a residuated lattice such that
are distributive:
Remark:
Morphisms
Let and be distributive residuated lattices. A
morphism from to is a function
that is a homomorphism:text/html2019-06-16T03:56:48-07:00Peter Jipsencommutative_residuated_lattices
http://math.chapman.edu/~jipsen/structures/doku.php/commutative_residuated_lattices?rev=1560682608&do=diff
Commutative residuated lattices
Abbreviation: CRL
Definition
A <b><i>commutative residuated lattice</i></b> is a residuated lattice such that
is commutative:
Remark:
Morphisms
Let and be commutative residuated lattices. A
morphism from to is a function
that is a homomorphism:text/html2019-06-15T06:34:32-07:00Peter Jipsencommutative_residuated_partially_ordered_monoids
http://math.chapman.edu/~jipsen/structures/doku.php/commutative_residuated_partially_ordered_monoids?rev=1560605672&do=diff
Commutative residuated partially ordered monoids
Abbreviation: CRPoMon
Definition
A <b><i>commutative residuated partially ordered monoid</i></b> is a residuated partially ordered monoid such that
is <b><i>commutative</i></b>:
Remark: This is a template.
If you know something about this class, click on the ``Edit text of this page'' link at the bottom and fill out this page.<b><i></i></b><b><i>Title</i></b><b>1</i></b>text/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>