# Terminology: Lexicographical order [closed]

I would like to order a group of things by a set of rules of decreasing precedence. Please critique this sentence to help illustrate that:

We define a lexicographical ordering for sheep by looking at the following properties in decreasing precedence:

• Age in years (rounded down), lower first
• Number of legs, higher first
• First name in dictionary order
• Average length of fur
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## closed as not a real question by Gjergji Zaimi, Robin Chapman, Loop Space, Yemon Choi, Anton GeraschenkoMay 12 '10 at 17:17

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Where is the problem/question? – Martin Brandenburg May 12 '10 at 0:09
The question seems to be, what's the terminology you'd use for specifying such an order? I'm not sure there is any short way of saying it, though; the above seems an OK description. – Harry Altman May 12 '10 at 8:04
There is a confusion between Lexicographical and Lexical. Among many other troubles, the first name entry is not a well defined entry because of the variable length in a lexicographical order where elements are concatenated. – ogerard May 13 '10 at 7:29

The lexical order of a well ordered list of orders $\langle X_\alpha,\leq_\alpha\rangle$ is the order on the product space $\Pi_\alpha X_\alpha$ placing $s$ before $t$ if $s_\alpha\lt_\alpha t_\alpha$ on the least difference coordinate.
Another difference is that you don't actually have orders, but pre-orders, since two sheep can have the same age, and so on. In this general setting, you have a set $X$ with a well-ordered sequence of pre-orders $\leq_\alpha$, and you define $a\leq b$ if the least index $\alpha$ for which $a$ and $b$ are not equivalent (if any such $\alpha$ exists) has $a\lt_\alpha b$.
It is important that the list of orders is well-ordered, since otherwise there may be no such least difference coordinate $\alpha$, and the idea will run aground. But any finite list of criteria is of course well-ordered, as is any $\omega$-sequence of orders.