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In fact, free complete lattices generally do not exist. Of course, one can formulate a word problem similar to the one for the case of lattices, but the collection of all possible words (or "terms") in this case would be a proper class, because arbitrary meets and joins comprise operations for argument sets of every cardinality.
In computational mathematics, a word problem is the problem of deciding whether two given expressions are equivalent with respect to a set of rewriting identities. A prototypical example is the word problem for groups , but there are many other instances as well.
Complete (complexity), a notion referring to a problem in computational complexity theory that all other problems in a class reduce to Turing complete set, a related notion from recursion theory; Completeness (knowledge bases), found in knowledge base theory
The word problem was one of the first examples of an unsolvable problem to be found not in mathematical logic or the theory of algorithms, but in one of the central branches of classical mathematics, algebra. As a result of its unsolvability, several other problems in combinatorial group theory have been shown to be unsolvable as well.
Semantic completeness is the converse of soundness for formal systems. A formal system is complete with respect to tautologousness or "semantically complete" when all its tautologies are theorems, whereas a formal system is "sound" when all theorems are tautologies (that is, they are semantically valid formulas: formulas that are true under every interpretation of the language of the system ...
Also note that the empty set usually has upper bounds (if the poset is non-empty) and thus a bounded-complete poset has a least element. One may also consider the subsets of a poset which are totally ordered, i.e. the chains. If all chains have a supremum, the order is called chain complete. Again, this concept is rarely needed in the dual form.
The Knuth–Bendix completion algorithm (named after Donald Knuth and Peter Bendix [1]) is a semi-decision [2] [3] algorithm for transforming a set of equations (over terms) into a confluent term rewriting system. When the algorithm succeeds, it effectively solves the word problem for the specified algebra.
Word problem from the Līlāvatī (12th century), with its English translation and solution. In science education, a word problem is a mathematical exercise (such as in a textbook, worksheet, or exam) where significant background information on the problem is presented in ordinary language rather than in mathematical notation.