MathStructures
http://mathv.chapman.edu/~jipsen/structures/
2018-08-14T03:56:11-07:00MathStructures
http://mathv.chapman.edu/~jipsen/structures/
http://mathv.chapman.edu/~jipsen/structures/lib/images/favicon.icotext/html2018-08-11T14:23:26-07:00Peter Jipsencommutative_bck-algebras
http://mathv.chapman.edu/~jipsen/structures/doku.php/commutative_bck-algebras?rev=1534022606&do=diff
Commutative BCK-algebras
Abbreviation: ComBCK
Definition
A <b><i>commutative BCK-algebra</i></b> is a structure of type such that
(1):
(2):
(3):
(4):
(5):
Remark:
Note that the commutativity does not refer to the operation , but rather to the
term operation , which turns out to be a meet with respect
to the following partial order:<b><i>commutative BCK-algebra</i></b>text/html2018-08-11T14:15:04-07:00Peter Jipsenbck-algebras
http://mathv.chapman.edu/~jipsen/structures/doku.php/bck-algebras?rev=1534022104&do=diff
BCK-algebras
Abbreviation: BCK
Definition
A <b><i>BCK-algebra</i></b> is a structure of type such that
(1):
(2):
(3):
(4):
Remark:
is a partial order, with as least element.
BCK-algebras provide algebraic semantics for BCK-logic, named after
the combinators B, C, and K by C. A. Meredith, see .<b><i>BCK-algebra</i></b><b><i>Formal logic</i></b><b><i>BCK-algebras do not form a variety</i></b><b>28</i></b><b><i>Interpolation and amalgamation properties of BCK-algebras</i></b…text/html2018-08-04T22:44:58-07:00Peter Jipsenseparation_algebras - created
http://mathv.chapman.edu/~jipsen/structures/doku.php/separation_algebras?rev=1533447898&do=diff
Separation algebras
Abbreviation: SepAlg
Definition
A <b><i>separation algebra</i></b> is a generalized separation algebra such that
is <b><i>commutative</i></b>: .
I.e., a separation algebra is a cancellative commutative partial monoid.
Morphisms
Let and be cancellative partial monoids. A morphism from to is a function that is a homomorphism:
and
if then .text/html2018-08-04T18:48:34-07:00Peter Jipsengeneralized_separation_algebras - created
http://mathv.chapman.edu/~jipsen/structures/doku.php/generalized_separation_algebras?rev=1533433714&do=diff
Generalized separation algebras
Abbreviation: GSepAlg
Definition
A <b><i>generalized separation algebra</i></b> is a cancellative partial monoid such that
is <b><i>conjugative</i></b>: .
Morphisms
Let and be cancellative partial monoids. A morphism from to is a function that is a homomorphism:
and
if then .text/html2018-08-04T18:39:19-07:00Peter Jipsenconjugative_binars - created
http://mathv.chapman.edu/~jipsen/structures/doku.php/conjugative_binars?rev=1533433159&do=diff
Conjugative binars
Abbreviation: ConBin
Definition
A <b><i>conjugative binar</i></b> is a binar such that
is conjugative: .
Morphisms
Let and be commutative binars. A morphism from to is a function that is a homomorphism:
Examples
Example 1:text/html2018-08-04T18:29:41-07:00Peter Jipsenindex.html
http://mathv.chapman.edu/~jipsen/structures/doku.php/index.html?rev=1533432581&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/html2018-08-04T18:24:11-07:00Peter Jipsencancellative_partial_monoids - created
http://mathv.chapman.edu/~jipsen/structures/doku.php/cancellative_partial_monoids?rev=1533432251&do=diff
Cancellative partial monoids
Abbreviation: CanPMon
Definition
A <b><i>cancellative partial monoid</i></b> is a partial monoid such that
is <b><i>left-cancellative</i></b>: implies and
is <b><i>right-cancellative</i></b>: implies .
Morphisms
Let and be cancellative partial monoids. A morphism from to is a function that is a homomorphism:
and
if then .text/html2018-08-04T18:07:47-07:00Peter Jipsenpartial_groupoids
http://mathv.chapman.edu/~jipsen/structures/doku.php/partial_groupoids?rev=1533431267&do=diff
Partial groupoids
Abbreviation: Pargoid
Definition
A <b><i>partial groupoid</i></b> is a structure , where
is a <b><i>partial binary operation</i></b>, i.e., .
Remark: The domain of definition of is Dom
Morphisms
Let and be partial groupoids. A morphism from to is a function that is a homomorphism:
if then <b><i>The theory of partial algebraic operations</i></b>text/html2018-08-04T17:55:44-07:00Peter Jipsenpartial_semigroups
http://mathv.chapman.edu/~jipsen/structures/doku.php/partial_semigroups?rev=1533430544&do=diff
Partial semigroups
Abbreviation: PSgrp
Definition
A <b><i>partial semigroup</i></b> is a structure , where
is a <b><i>partial binary operation</i></b>, i.e., and
is <b><i>associative</i></b>: or imply .
Morphisms
Let and be partial groupoids. A morphism from to is a function that is a homomorphism:
if thentext/html2018-08-04T17:19:49-07:00Peter Jipsenright_quasigroups
http://mathv.chapman.edu/~jipsen/structures/doku.php/right_quasigroups?rev=1533428389&do=diff
Right quasigroups
Abbreviation: RQgrp
Definition
A <b><i>right quasigroup</i></b> is a structure of type such that
Remark:
Morphisms
Let and be right quasigroups. A morphism from to is a function that is a homomorphism:
, .
Examples
Example 1:text/html2018-08-04T15:53:48-07:00Peter Jipsenright_hoops - created
http://mathv.chapman.edu/~jipsen/structures/doku.php/right_hoops?rev=1533423228&do=diff
Right hoops
Definition
A <b><i>right hoop</i></b> is a structure of type such that
is a monoid
Remark:
This definition shows that right hoops form a variety.
Right hoops are partially ordered by the relation .
The operation is a meet with respect to this order.<b><i>right hoop</i></b><b><i>right hoop</i></b>text/html2018-08-04T15:39:27-07:00Peter Jipsenhoops
http://mathv.chapman.edu/~jipsen/structures/doku.php/hoops?rev=1533422367&do=diff
Hoops
Definition
A <b><i>hoop</i></b> is a structure of type such that
is a commutative monoid
Remark:
This definition shows that hoops form a variety.
Hoops are partially ordered by the relation .
The operation is a meet with
respect to this order.<b><i>hoop</i></b><b><i>hoop</i></b>text/html2018-08-01T11:42:55-07:00Peter Jipsendistributive_lattices
http://mathv.chapman.edu/~jipsen/structures/doku.php/distributive_lattices?rev=1533148975&do=diff
Distributive lattices
Abbreviation: DLat
Definition
A <b><i>distributive lattice</i></b> is a lattice such that
distributes over :
Definition
A <b><i>distributive lattice</i></b> is a lattice such that
distributes over :
Definition
A <b><i>distributive lattice</i></b> is a lattice such that<b><i>distributive lattice</i></b><b><i>distributive lattice</i></b><b><i>On the number of distributive lattices</i></b><b><i>Postulates for distributive lattices</i></b>