In normal form, they read as follows:

$$\begin{array}{l}{{\overline{{{p}}}_{x y}\lor\ \overline{{{p}}}_{y z}\lor\ p_{x z},}}\\ {{\overline{{{p}}}_{x y}\lor\ \overline{{{p}}}_{y x},}}\\ {{p_{x y}\lor\ p_{y x},}}\end{array}$$
$${\mathrm{for~any~pairwise~different~}}x,y,z\in A;$$
$${\mathrm{~for~any~two~different~}}x,y\in A;$$ $${\mathrm{~for~any~two~different~}}x,y\in A,$$
pxy ∨ pyz ∨ pxz, for any pairwise different *x, y, z* ∈ A; (65)
pxy ∨ pyx, for any two different *x, y* ∈ A; (66)
pxy ∨ pyx, for any two different *x, y* ∈ A, (67)
where the last one does not have a Horn character.

Similarly to the case of an equivalence relation, this doctrine is also disjoint–resolvable and unquestionable. Its Blake canonical form, with the *tertium non datur* clauses included, consists of all clauses of the form

$$\overline{{{p}}}_{x_{0}x_{1}}\ \vee\ \overline{{{p}}}_{x_{1}x_{2}}\ \vee\ \ldots\ \vee\ \overline{{{p}}}_{x_{n-1}x_{n}}\ \vee\ p_{x_{0}x_{n}},$$
$$(68)$$

, (68)
where x0x1 *. . . x*n is a non-cyclic path, *together with* all clauses that are obtained from (68) by replacing one or more pxixi+1 by pxi+1xi
, and/or replacing px0xn by pxnx0
(i. e. by identifying pxy with pyx for any two different x, y ∈ A).

Let us remark also that applying Theorem 4.10 to this domain gives the finite case of the well-known theorem of Edward Szpilrajn about the extendability of partial orders to total orders [26].

In this case, our general method corresponds essentially to the voting method introduced in 1997 by Markus Schulze [24 , 25 ; 27 : p. 228–232], often called the method of paths. More precisely, when the original valuation is balanced —which happens when every voter expresses a comparison (a preference or a tie) about each pair of options— we coincide with Schulze's method except for his subsequent treatment of indecisions; when the preferential information given by the voters is not complete, this procedure does not coincide exactly with any of the variants that Schulze gives in [25], but it has the same spirit. For more details, we refer the reader to [*4, 5*], where it is shown that this procedure can be extended to a continuous rating method that allows to sense the closeness of two candidates at the same time that it allows to recognise certain situations that are quite opposite to a tie.

Here we will only draw attention to significance of certain autarkic sets of this doctrine. In fact, one can easily check the autarkic character of any set of the form Σ = { pxy | x ∈ *X, y* ∈ A \ X }, where X is any proper subset of A. Applied to such sets, Theorem 4.15 ensures that the method of paths satisfies the following majority property [4 : Thm. 10.1; 5 : Thm. 8.1]:
If for each member of X and every member of A \ X there are more than half of the individual votes where the former is preferred to the latter, then the resulting social ranking also prefers each member of X to every member of A \ X .