chains - MathStructures

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=====Chains=====

====Definition====
A \emph{chain} is a [[partially ordered set]] $\mathbf{C}=\langle C,\le\rangle$ such that


$\le$ is a total order:  $x\le y \mbox{ or } y\le x$


Remark: 

==Morphisms==
Let $\mathbf{C}$ and $\mathbf{D}$ be chains. A morphism from $\mathbf{C}$ to $\mathbf{D}$ is a function $h:C\rightarrow D$ that is a orderpreserving: 

$x\le y\Longrightarrow h(x)\le h(y)$

====Examples====
Example 1: 

====Basic results====


====Properties====
^[[Classtype]]  |Universal |
^[[Quasiequational theory]]  | |
^[[First-order theory]]  | |
^[[Amalgamation property]]  | |
^[[Strong amalgamation property]]  | |
^[[Epimorphisms are surjective]]  | |
====Finite members====

$\begin{array}{lr}
f(1)= &1\\
f(2)= &1\\
f(3)= &1\\
f(4)= &1\\
f(5)= &1\\
f(6)= &1\\
\end{array}$

====Subclasses====
[[Well-ordered chains]] 

[[Dense linear orders]] 

====Superclasses====
[[Trees]] 


====References====

[(Ln19xx>
)]

Trace: • integral_residuated_lattices • finite_monoids • congruence_e-regular • weakly_associative_relation_algebras • chains • pseudocomplemented_distributive_lattices • metric_spaces • lattice-ordered_semigroups • directed_graphs • boolean_algebras_with_operators