# Differences

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+ | =====Preordered sets===== | ||

+ | Abbreviation: **Qoset** | ||

+ | |||

+ | ====Definition==== | ||

+ | A \emph{preordered set} (also called a \emph{quasi-ordered set} or \emph{qoset} for short) is a structure $\mathbf{P}=\langle P,\preceq\rangle$ | ||

+ | such that $P$ is a set and $\preceq $ is a binary relation on $P$ that is | ||

+ | |||

+ | reflexive: $x\preceq x$ and | ||

+ | |||

+ | transitive: $x\preceq y \text{ and } y\preceq z\Longrightarrow x\preceq z$ | ||

+ | |||

+ | Remark: | ||

+ | |||

+ | ==Morphisms== | ||

+ | Let $\mathbf{P}$ and $\mathbf{Q}$ be qosets. A morphism from $\mathbf{P}$ to | ||

+ | $\mathbf{Q}$ is a function $f:P\rightarrow Q$ that is preorder-preserving: | ||

+ | |||

+ | $x\preceq y\Longrightarrow f(x)\preceq f(y)$ | ||

+ | |||

+ | ====Examples==== | ||

+ | Example 1: | ||

+ | |||

+ | ====Basic results==== | ||

+ | |||

+ | |||

+ | ====Properties==== | ||

+ | ^[[Classtype]] |Universal Horn class | | ||

+ | ^[[Universal theory]] |Decidable | | ||

+ | ^[[First-order theory]] |Undecidable | | ||

+ | ^[[Amalgamation property]] | | | ||

+ | ^[[Strong amalgamation property]] | | | ||

+ | ^[[Epimorphisms are surjective]] | | | ||

+ | ====Finite members==== | ||

+ | |||

+ | $\begin{array}{lr} | ||

+ | f(1)= &1\\ | ||

+ | f(2)= &2\\ | ||

+ | f(3)= &\\ | ||

+ | f(4)= &\\ | ||

+ | f(5)= &\\ | ||

+ | f(6)= &\\ | ||

+ | f(7)= &\\ | ||

+ | \end{array}$ | ||

+ | |||

+ | ====Subclasses==== | ||

+ | [[Posets]] | ||

+ | |||

+ | [[Connected qosets]] | ||

+ | |||

+ | ====Superclasses==== | ||

+ | [[Binary relational structures]] | ||

+ | |||

+ | |||

+ | ====References==== | ||

+ | |||

+ | [(Ln19xx> | ||

+ | )] |

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