Definitions of greatest and least elements in terms of strict orderings

[hmb]

Homework Statement

State the definitions of greatest and least elements in terms of strict orderings.

Homework Equations

Let $$\leq$$ be an ordering of $$A$$ and $$<$$ be a strict ordering on $$A$$, and let $$B \subseteq A$$.

$$b \in B$$ is the greatest element of $$B$$ in the ordering $$\leq$$ if, for every $$x \in B$$, $$x \leq b$$.

$$b \in B$$ is the least element of $$B$$ in the ordering $$\leq$$ if, for every $$x \in B$$, $$b \leq x$$.

The Attempt at a Solution

$$b \in B$$ is the greatest element of $$B$$ in the ordering $$<$$ if, for every $$x \in B$$, $$x < b$$.

But then there is no greatest element, because $$x < b$$ implies $$x \neq b$$. So maybe it should be:

$$b \in B$$ is the greatest element of $$B$$ in the ordering $$<$$ if, for every $$x \in B$$ and $$x \neq b$$, $$x < b$$?

$$b \in B$$ is the least element of $$B$$ in the ordering $$<$$ if, for every $$x \in B$$, $$b < x$$.

But then there is no least element, because $$x < b$$ implies $$x \neq b$$. So maybe it should be:

$$b \in B$$ is the least element of $$B$$ in the ordering $$<$$ if, for every $$x \in B$$ and $$x \neq b$$, $$b < x$$?

While I am at it I might as well also check that I've got some other definitions right:

$$b \in B$$ is a maximal element of $$B$$ in the ordering $$<$$ if there exists no $$x \in B$$ such that $$b < x$$.

$$b \in B$$ is a minimal element of $$B$$ in the ordering $$<$$ if there exists no $$x \in B$$ such that $$x < b$$.

$$a \in A$$ is an upper bound of $$B$$ in the ordered set $$(A, <)$$ if $$x < a$$ for all $$a \in B$$.

$$a \in A$$ is called a supremum of $$B$$ in $$(A, <)$$ if it is the least element of the set of all upper bounds of $$B$$ in $$(A, <)$$.

$$a \in A$$ is a lower bound of $$B$$ in the ordered set $$(A, <)$$ if $$a < x$$ for all $$x \in B$$.

$$a \in A$$ is called an infimum of $$B$$ in $$(A, <)$$ if it is the greatest element of the set of all lower bounds of $$B$$ in $$(A, <)$$.

Let $$a, b \in A$$, and let $$<$$ be an ordering of $$A$$. We say that $$a$$ and $$b$$ are comparable in the ordering $$<$$ if $$a < b$$ or $$b < a$$. We say that $$a$$ and $$b$$ are incomparable if they are not comparable (i.e., if $$a \neq b$$ and neither $$a < b$$ nor $$b < a$$ .

 

[micromass]

$$b \in B$$ is the greatest element of $$B$$ in the ordering $$<$$ if, for every $$x \in B$$ and $$x \neq b$$, $$x < b$$?

This is correct.

 

[hmb]

Great, thank you for your help. I will take it that the corresponding definition of "least element" is correct as well then.

Thanks again.

 

[micromass]

Yes, the other definitions are correct as well