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Wikipedia says following about lexeme:
A lexeme is a unit of lexical meaning... For example, in English, run, runs, ran and running are forms of the same lexeme, which can be represented as
Wikipedia says following about stem:
In linguistics, a stem is a part of a word responsible for its lexical meaning.
Am unable to understand whether both terms mean the same.
A "lexeme" is a theoretical thing, a unit in the mental lexicon. You can think of it as being an entire dictionary entry, but in our mental knowledge bank of what words mean rather than a physical book.
A "stem" is a practical thing: it's the part of a word that you stick affixes onto. The stem of play, playing, plays, played, etc is play-. In English the stem usually looks like an actual word, but it doesn't have to be: in Latin, the root of the Latin words amīcus, amīcī, amīcum, amīcō, etc is amīc-, which isn't a valid Latin word on its own. So you'll sometimes find the word "lemma" used to mean "the stem, with some default affix attached to make it a real word" (in Latin, that would be amīcus).
The concept of a lexeme is pretty standard across languages. No matter what language you speak, you have some sort of mental understanding of what words mean. But the concept of a stem is very useful in some languages and nigh useless in others. It all depends how much the language uses affixes.
From the computational linguistics point of view, and especially targeting moderately inflected languages, a lexeme can be thought as a set of all inflected forms of a word. Such a set - a lexeme - can be represented by something uniquely describing it, which is usually a lemma, a basic form of the word (this is quite language dependent). The stem can be thought as a "poor man's lemma", a part of the word that remains unchanged (there are exceptions, though - e.g. if the root morpheme mutates predictably in some cases, stemming can "undo" this mutation) and in some circumstances it could be used to represent a lexeme (but usually isn't).
Stemming is usually performed by a relatively simple algorithm without deeper linguistic consideration (such as meaning, homonymy etc.) and used for full text searches and such.
So, in English (a bad language to base an example on), a set {run, ran, running, runs}, usually with a connection to its meaning, and/or part of speech (thus depending on your level of abstraction, "run" as a noun can be a different lexeme {run, runs}) is a lexeme. A natural way of representing the lexeme is the lemma "run"; your stemming algorithm will likely give you the "run" stem as well, but it might fail for the "ran" word form.
Lexemes and stems are not related.
A lexeme is an abstract concept that represents the core meaning shared by a set of closely-related word-forms (e.g. the concept of 'live' is the lexeme for the word-forms 'lives', 'live', and 'lived').
Lexemes are abstract concepts without phonological forms; so the word 'live' is not a lexeme, but the abstract concept of 'live' is. To make this distinction, lexemes are typically represented by writing its lemma (i.e. dictionary or citation form) in small caps, like this: ʟɪᴠᴇ.
A stem is the part of a word that affixes attaches to. For example, in the word 'readable', 'read' is the stem and '-able' is the suffix. In the word 'readability', 'readable' is the stem and '-ity' is the suffix.
