MathStructures
http://mathv.chapman.edu/~jipsen/structures/
2018-02-23T10:40:32-08:00MathStructures
http://mathv.chapman.edu/~jipsen/structures/
http://mathv.chapman.edu/~jipsen/structures/lib/images/favicon.icotext/html2017-12-10T14:18:39-08:00Peter Jipsenindex.html
http://mathv.chapman.edu/~jipsen/structures/doku.php/index.html?rev=1512944319&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/html2017-10-02T10:57:02-08:00Peter Jipsenequations
http://mathv.chapman.edu/~jipsen/structures/doku.php/equations?rev=1506967022&do=diff
Syntax | Terms | Equations | Horn formulas | Universal formulas | First-order formulas | Theories
Here we list equations, with the shorter term on the right (if possible).
1 trivial equations: one-element algebras 2 identity operation: 3 involutive operation: 4 inverse operations: 5 inside absorption: 6 outside absorption: 7 order- operation: 8 -idempotent 9 constant operations: 10 left projection: right projection: 11 idempo…text/html2017-10-01T20:07:06-08:00Peter Jipsenhorn_formulas
http://mathv.chapman.edu/~jipsen/structures/doku.php/horn_formulas?rev=1506913626&do=diff
Syntax | Terms | Equations | Horn formulas | Universal formulas | First-order formulas | Theories
A list of atomic formulas, quasiequations and universal Horn formulas.
reflexive: transitive: antisymmetric: left cancellation: right cancellation: left R-preserving: right R-preserving:text/html2017-08-05T13:07:41-08:00Peter Jipsensqrt-quasi-mv-algebras
http://mathv.chapman.edu/~jipsen/structures/doku.php/sqrt-quasi-mv-algebras?rev=1501963661&do=diff
Sqrt-quasi-MV-algebras
Abbreviation: sqMV
Definition
A <b><i>quasi-MV-algebra</i></b> is a
structure such that is a unary operation,
is a quasi-MV-algebra,
,
, and
.
Morphisms
Let and be qMV-algebras. A morphism from to is a function that is a homomorphism: <b><i>Expanding quasi-MV algebras by a quantum operator</i></b><b>87</i></b>text/html2017-08-05T13:04:54-08:00Peter Jipsenquantales
http://mathv.chapman.edu/~jipsen/structures/doku.php/quantales?rev=1501963494&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/html2017-08-03T14:44:29-08:00Peter Jipsensemidistributive_lattices
http://mathv.chapman.edu/~jipsen/structures/doku.php/semidistributive_lattices?rev=1501796669&do=diff
Semidistributive lattices
Abbreviation: SdLat
Definition
A <b><i>semidistributive lattice</i></b> is a lattice such that
SD:
SD:
Morphisms
Let and be semidistributive lattices. A morphism from to is a function
that is a homomorphism:text/html2017-08-03T14:42:25-08:00Peter Jipsenjoin-semidistributive_lattices
http://mathv.chapman.edu/~jipsen/structures/doku.php/join-semidistributive_lattices?rev=1501796545&do=diff
Join-semidistributive lattices
Abbreviation: JsdLat
Definition
A <b><i>join-semidistributive lattice</i></b> is a lattice that satisfies
the join-semidistributive law SD:
Morphisms
Let and be join-semidistributive lattices. A morphism from to is a function
that is a homomorphism:text/html2017-08-03T14:41:29-08:00Peter Jipsenmeet-semidistributive_lattices
http://mathv.chapman.edu/~jipsen/structures/doku.php/meet-semidistributive_lattices?rev=1501796489&do=diff
Meet-semidistributive lattices
Abbreviation: MsdLat
Definition
A <b><i>meet-semidistributive lattice</i></b> is a lattice that satisfies
the meet-semidistributive law SD:
Morphisms
Let and be meet-semidistributive lattices. A morphism from to is a function
that is a homomorphism:text/html2017-02-12T07:27:32-08:00Peter Jipsengoedel_algebras
http://mathv.chapman.edu/~jipsen/structures/doku.php/goedel_algebras?rev=1486913252&do=diff
Gödel algebras
Abbreviation: GödA
Definition
A <b><i>Gödel algebra</i></b> is a Heyting algebras such that
Remark:
Gödel algebras are also called <b><i>linear Heyting algebras</i></b> since subdirectly irreducible Gödel algebras are linearly ordered Heyting algebras.<b><i>Gödel algebra</i></b>text/html2016-12-09T19:25:15-08:00Peter Jipsendistributive_residuated_lattices
http://mathv.chapman.edu/~jipsen/structures/doku.php/distributive_residuated_lattices?rev=1481340315&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/html2016-12-02T08:34:07-08:00Peter Jipseninverse_semigroups
http://mathv.chapman.edu/~jipsen/structures/doku.php/inverse_semigroups?rev=1480696447&do=diff
Inverse semigroups
Abbreviation: InvSgrp
Definition
An <b><i>inverse semigroup</i></b> is a structure such that
is associative:
is an inverse: and
idempotents commute:
Morphisms
Let and be inverse semigroups. A morphism from
to is a function that is a
homomorphism: <b><i>symmetric inverse semigroup</i></b>