(a) describes transitivity (it requires $Pxz$ exactly in those cases when $Pxy$ and $Pyz$ ). (b) describes antisymmetry (it requires $x=y$ exactly in those cases when $Pxy$ and $Pyx$ ). (c) describes the case when if for each element there is a “greater” one ($\forall x\exists yPxy$ ) then there is a maximal element ($\exists y\forall xPxy$ ).

We consider the minimal language required in our case, $(=,\forall,P)$ , and structures $\mathfrak{A}$ such that $|\mathfrak{A}|=\{a,b\}$ . We still need to specify the relation $P$ on $|\mathfrak{A}|$ . There are 16 possibilities, 10 being essentially different (there are no automorphisms between any two of them).

$P$	(a)	(b)	(c)
$\emptyset$	+	+	+
$\{<a,a>\}$	+	+	+
$\{<a,b>\}$	+	+	+
$\{<a,a>,<a,b>\}$	+	+	+
$\{<a,a>,<b,a>\}$	+	+	+
$\{<a,a>,<b,b>\}$	+	+
$\{<a,b>,<b,a>\}$
$\{<a,a>,<a,b>,<b,a>\}$			+
$\{<a,a>,<a,b>,<b,b>\}$	+	+	+
$\{<a,a>,<a,b>,<b,a>,<b,b>\}$	+		+

Therefore, we already have two examples needed: the relation $\{<a,a>,<b,b>\}$ (which can be also described simply as $x=y$ ) is transitive antisymmetric, and for each $x$ there is $y=x$ such that $Pxy$ , but there is no maximal element; and the relation $\{<a,a>,<a,b>,<b,a>,<b,b>\}$ (which can be also described as the complete one when for every $x$ and $y$ , $Pxy$ ) is transitive with a maximal element (in fact, two maximal elements) but not antisymmetric ($Pab$ and $Pba$ but $a\neq b$ ).

We still need an example of the structure $\mathfrak{A}$ and relation $P$ on $|\mathfrak{A}|$ such that $P$ is not transitive, but it is antisymmetric and either not every element has a greater element or there is a maximal element. For this, we obviously need at least three elements. Indeed, if $P$ is not transitive, then for some $x$ , $y$ and $z$ , $Pxy$ and $Pyz$ but not $Pxz$ , therefore, $x\neq y\neq z$ . But if $x=z$ , then $P$ is not antisymmetric. But the relation $P=\{<a,b>,<b,c>\}$ on $|\mathfrak{A}|=\{a,b,c\}$ already satisfies (b) and (c) but not (a), as it is not transitive ($Pab$ and $Pbc$ but not $Pac$ ), but it is antisymmetric (there are no different $x$ and $y$ such that $Pxy$ and $Pyx$ , in fact, there are no such $x$ and $y$ at all) and also not for every $x$ there is $y$ such that $Pxy$ (there is no for $x=c$ ), i.e. $\not\vDash_{\mathfrak{A}}\forall x\exists yPxy$ and $\vDash_{\mathfrak{A}}(\forall x\exists yPxy\rightarrow\exists y\forall xPxy)$ .