There are more than 300 different definitions of the sentence, but not a universal one

32. One-member or Eliptical Sentences
On the basis of the representation of the subject group and the predicate group in the outer structure of the sentence, sentences are subdivided into complete sentences and incomplete sentences: in complete sentences both the subject group and the predicate group are present; they are also called “two-member sentences” or “two-axis sentences”; if only one axis is expressed in the outer structure of the sentence, the sentence is defined as incomplete; it is also called “one-member sentence”, “one-axis sentence”.
Traditionally, one-axis sentences and elliptical sentences are distinguished in the following way: only those sentences in which the nominative parts are contextually omitted are considered to be elliptical, e.g.:Who is there? – Your brother. Since the missing parts are easily restored (“understood”) from the context, elliptical sentences are treated as two-member sentences. 
One-member sentences are traditionally treated as those which do not imply the missing member on contextual lines, e.g.: What a nice day! 
But, strictly speaking, “ellipsis”, as a mechanism of sentence structure-curtailing, can affect both two-axis sentences and one-axis sentences
) complete & elliptical (a sentence which has 1 or both principal members are omitted but can easily restored)
Профессор бархударов considers sent-s of the type “come” elliptical ( not one-member) as they can be traced to full sen-s ( “you come”)
Ellipsis: initial (what’s her name-don’t know), medial (you sure I ‘ve done?) final (who is coming-tom &mary)
Ellipses: textual(in written English) situational( in spoken)
There is a speсial kind of ellipses - Representation – a variety of substitution, when a part is used instead of the whole. It is used to avoid repetition without destroying structural and semantic unity. Representatives are always correlative because a represanter is a part of what it replaces. Modal, auxil. Verbs, link verbs, the part of the infinitive, the negat. Particle “not” (Ex. – Will you come? – yes, I’d like to; - Will you come? – I think no)
1-member sentence is a sentence in which there’s 1 principle part & the other part cannot be supplied without a change in the structure.
The problems.

1. 
   1-m. sentence (complete, 1 sentence member, self-sufficent)
   

Whether 1 member is a separate structure type or is it an ellipsis of 2 member sentence. How to name the only principal sen-ce part. Some say we cant name the subject or the predicate because they are correlative members.
H.Sweet. 1-m. sentence – sen-ce expressed by the words “yes/no”