Properties
The cancellation property states that equivalence is maintained under right cancellation. That is, if, then . Here, the notation denotes right cancellation, the removal of the first occurrence of the letter a from the string w, starting from the right-hand side. Equivalence is also maintained by left-cancellation. Several corollaries follow:
- Embedding: if and only if for strings x and y. Thus, the trace monoid is a syntactic monoid.
- Independence: if and, then a is independent of b. That is, . Furthermore, there exists a string w such that and .
- Projection rule: equivalence is maintained under string projection, so that if, then .
A strong form of Levi's lemma holds for traces. Specifically, if for strings u, v, x, y, then there exist strings and such that for all letters and such that occurs in and occurs in, and
Read more about this topic: Trace Monoid
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)