Differences

This shows you the differences between two versions of the page.

partially_ordered_semigroups [2010/07/29 15:46]
127.0.0.1 external edit
partially_ordered_semigroups [2018/05/06 07:33] (current)
jipsen
Line 11: Line 11:
$\cdot$ is \emph{orderpreserving}:  $x\le y\Longrightarrow xz\le yz \text{ and } zx\le zy$ $\cdot$ is \emph{orderpreserving}:  $x\le y\Longrightarrow xz\le yz \text{ and } zx\le zy$
- 
-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. 
- 
-It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes. 
==Morphisms== ==Morphisms==
Line 21: Line 16:
$h(x \cdot y)=h(x) \cdot h(y)$, $h(x \cdot y)=h(x) \cdot h(y)$,
$x\le y\Longrightarrow h(x)\le h(y)$ $x\le y\Longrightarrow h(x)\le h(y)$
- 
-====Definition==== 
-A \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle 
-...\rangle$ such that 
- 
-$...$ is ...:  $axiom$ 
-   
-$...$ is ...:  $axiom$ 
====Examples==== ====Examples====
-Example 1: +Example 1: The natural numbers larger than 1, with addition, or with multiplication.
====Basic results==== ====Basic results====
Line 37: Line 24:
====Properties==== ====Properties====
-Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described. 
^[[Classtype]]                        |quasivariety  | ^[[Classtype]]                        |quasivariety  |
Line 61: Line 47:
$\begin{array}{lr} $\begin{array}{lr}
  f(1)= &1\\   f(1)= &1\\
-  f(2)= &\\ +  f(2)= &11\\ 
-  f(3)= &\\+  f(3)= &173\\
  f(4)= &\\   f(4)= &\\
  f(5)= &\\   f(5)= &\\
\end{array}$     \end{array}$    
-$\begin{array}{lr} + 
-  f(6)= &\\ +Gajdos Kuril 2014 Ordered semigroups of size at most 7 and linearly ordered semigroups of size at most 10, Semigroup Forum 
-  f(7)= &\\ + 
-  f(8)= &\\ +Number of elements 5 6 7, values below are for partially ordered semigroups. 
-  f(9)= &\\ + 
-  f(10)= &\\ +Semigroups 198838 13457454 4207546916 
-\end{array}$+ 
 +Commutative semigroups 37248 1337698 71748346 
 + 
 +Monoids 13371 504634 32113642 
 + 
 +Bands 20305 494848 14349957 
 + 
 +Regular semigroups 22419 546386 15842224 
 + 
 +Inverse semigroups 2886 44275 830584 
 + 
 +2-nilpotent semigroups 243 1533 12038 
 + 
 +3-nilpotent semigroups 14150 2561653 3215028097
====Subclasses==== ====Subclasses====
-  [[Commutative partially ordered semigroups]]+[[Commutative partially ordered semigroups]]
-  [[Lattice-ordered semigroups]] expanded type+[[Lattice-ordered semigroups]] expanded type
====Superclasses==== ====Superclasses====
-  [[Partially ordered groupoids]]+[[Partially ordered groupoids]]
Line 90: Line 89:
F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 [[MRreview]] F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 [[MRreview]]
)] )]
-