Important characteristics of Euclidean time include the seven properties of immediacy, universality, continuity, staticity, unidirectionality, infinity, and linearity.
Time has a special and mysterious quality about it having to do with it occurring now, which we may name immediacy. Of an infinity of times, only the immediate instant is now. But the referent of now is constantly changing, so that it points to some different time, for example, now. What is constant about this ever-changing temporal pointer? Not the things that occur in time, which includes just about everything. Depending on our point of view we may call these things ``states'', or ``events'', or ``elements'' or even ``features'' or ``feature-values''.
In considering the mystery of immediacy, one may notice that its special, elusive character is not so special from an extra-temporal perspective (as in the later examination of a record of any particular temporal sequence). No event in the record is singled out as in any way different from any other event in the record, except that each occurs at its own unique time. The only remnant of now-ness left in the record is the property of each event that it occurs at its own individual immediate time, which is now relative to the event itself. This could be taken as evidence that records of temporal sequences have not recorded the essence of this principle of immediacy. However, for the purposes of examining records of events, immediacy is the principle that events occur ``at'' particular times; that is, that times are associated with events.
The other properties require less discussion. Universality is the principle that everything (in the universe under discussion) occurs at some time. Continuity in real time may be axiomatized as the statement that between every two (distinct) times there is another time. Staticity is the principle that an event (in this context, more properly called a ``state'') may extend across (be associated with) a contiguous sequence of times. Infinity is the principle that time does not begin or end.A.1Unidirectionality specifies not just that the two directions are distinct from each other (this follows from linearity) but that one direction is special, in that certain sequences of events may not be reversed; they can not in principle occur in the reverse order.
Linearity is defined in any mathematics-for-linguists text. The important properties here are that for a given element e in a sequence,
A: all other distinct elements either precede or followA.2 e.
B: no element may both precede and follow e.
We may define classes of time other than Euclidean time by eliminating certain of these properties. Thus discrete time drops the continuity principle, but retains infinity, linearity, immediacy, universality, staticity, and unidirectionality. Finite time, of which linguistic time is an instance, further eliminates the requirement of infinity. One might also drop the requirement of staticity, so that events have duration of only one time unit.
Linear, linguistic representations have the character of non-static, finite, discrete time. Thus, for example, linear sequences of feature matrices, which that form phonological representations in SPE, have the properties of immediacy (the association of elements with particular locations in the ordering), universality (the claim that all elements are thus associated with some location), and linearity (the complete ordering of all the locations), but not of continuity, infinity, or staticity. StaticityA.3 deserves special treatment in this case: geminate segments might be considered as states, since the same feature matrix repeats itself at two successive times. However, I take it as essential about SPE representations that repeating identical matrices are precisely that: repetitions; they are NOT the same matrix extended across two times. In this respect, SPE is unlike both its predecessors and its successors.
The linearity of time gives an order to temporal events/states/elementsA.4. In static finite-temporal representations, this ordering is represented by in some way adding to the description of each element of the sequence a uniquely ordered index. This is the implicit effect of writing forms in a sequence on paper: the physical layout provides an analog of the temporally ordered indexing of all the elements in the string. If this implicit ordering is done explicitly, by numbering the elements with an (ordered) index, then the actual location of elements in a particular physical representation becomes immaterial. Words in a sentence, for example, could be written on separate pieces of paper with their respective indices and tossed into the air: their linear ordering is still fully specified since their (ordered) indices are specified. It makes no difference whether elements are concretely put in order (for example, by printing them in a spatial sequence on a teletype machine) or ordered abstractly by associating unique indices/times - for which the set of indices has the properties of finite time - with the elements (for example, by storing numbers with the elements in unordered, random-access, computer memory). In either case their ordering is fully specified; that is, concrete and abstract ordering are equivalent for our purposes.
This view of ordering adds no new descriptive machinery to linguistic representations. The features of finite time are implicit in any finite, linearly ordered, non-reversible sequence of discrete elements. Sequences of linguistic elements may therefore be described in terms of discrete time. If one wishes to add to a system of linguistic representation explicit machinery which indexes each element with a distinct ``time'' (or number, say), this makes no further claims than those already implicitly made by the unindexed, concrete representation of the same sequences. Thus this view merely makes explicit and formal the formerly implicit and informal machinery which has always been used in linguistic representations.