A lexeme is a basic unit of meaning in a language. A lexeme can be a single word or set of words; any unit that depends upon all its morphological components to be understood qualifies. In other words, a lexeme is a unit of language that can no longer be understood when broken down into its component parts. This means that any inflected forms of a lexeme fall under the umbrella of that lexeme. Any verb, for example walk, is a single lexeme composed of a set of inflected forms, i.e. walking, walks, walked, etc. In addition, walker and walkers are not inflected forms of the lexeme walk, but forms rather of the lexeme walker; this is due to the difference in word class – walker is a noun, rather than a verb.
Phrasal verbs and compound words are also usually single lexemes, though they are composed of more than one grammatical word. For example, the word babysitter is a lexeme in the English language – a compound word, composed of baby and sitter, whose meaning cannot be divined by looking at its component parts. In addition, phrasal verbs like turn off or come back are also single lexemes – their component parts form a new, single meaning when combined.
Name and idioms, as well, are single lexemes – when someone mentions Sherlock Holmes, this name conjures up one single referent; thus there is a single meaning behind the two names. If you were to call someone a bad apple, you would be using a single lexeme to refer to this person. The phrase may be two words, but they are combined in such a context they have a single meaning. Conversely, if you were to see a bad apple among the fresh, shiny pile at the grocery store, this conventional connotation would be gone, and we would have two separate lexemes.
So to address the need for example sentences, you can’t really go wrong with any sentence at all in the English language, for languages are made up entirely of lexemes. A good sentence would perhaps be something along the lines of the following:
My mother told the babysitter not to come back after the teenager horsed around on the job.
Here we have babysitter, come back, teenager, and horse around as good examples of compound words and phrases as single lexemes, as well as the obvious mother and tell, and the phrase on the job.