Abstract Algebra

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Set Theory: Definition of a Total Ordering, Identification


A total order or total ordering on a set $S$ is a partial order $\preceq$ that satisfies the additional property, often called Comparability:

If $x,y\in S$, then either $x\preceq y$ or $y\preceq x$.

Which of the following is a total order?


Inclusion on the set of subsets of $\{1,2,3\}$


Lexicographic order on $P\times P$ where $P$ is the set of subsets of $\{0,1\}$ ordered by inclusion


Lexicographic ordering on $A\times B$, where $A$ is the set $\{2,6,30\}$ ordered by divisibility ($a\preceq b$ if and only if $a\mid b$), and $B$
is the set of integers with the usual ordering ($r\le s$ if and only if $s-r$ is a non-negative integer)


The ordering completely described by $a\preceq a$, $b\preceq b$ and $c\preceq c$ on the set $S=\{a,b,c\}$, where $a$, $b$, $c$ are distinct ($|S|=3$).


The Product order on $A\times A$, where $A=\{1,2,3\}$ ordered by the usual $\le$. Here, the product ordering is $(a_1,a_2)\le (a_1',a_2')$ if and only if $a_1\le a_1'$ and $a_2\le a_2'$