The equational theory of the implication-free fragment of action algebras, those equations not containing ?! or ?!, can be shown to coincide with the equational theory of Kleene algebras, also known as the regular expression equations.
12.
The equational theory of the implication-free fragment of action algebras, those equations not containing ?! or ?!, can be shown to coincide with the equational theory of Kleene algebras, also known as the regular expression equations.
13.
Now Birkhoff's HSP theorem for varieties can be stated as, every class of models of the equational theory of a class " C " of algebras is the Homomorphic image of a Subalgebra of a variety of Boolean algebras.
14.
In this way when talking about different equational theories we can push the rules to one side as being independent of the particular theories, and confine attention to the axioms as the only part of the axiom system characterizing the particular equational theory at hand.
15.
In this way when talking about different equational theories we can push the rules to one side as being independent of the particular theories, and confine attention to the axioms as the only part of the axiom system characterizing the particular equational theory at hand.
16.
When the commutative diagrams of a category are interpreted as a typed equational theory whose objects are the types, a posetal category corresponds to an inconsistent theory understood as one satisfying the axiom " x " = " y " at all types.
17.
In 1998, Mitchell Wand showed that adding a fexpr device to lambda calculus & mdash; a device that suppresses rewriting of operands & mdash; produces a formal system with a trivial equational theory, rendering it impossible to make source-to-source optimizations without a whole-program analysis.
18.
A model of the equational theory of the algebra of " all " operations on { 0, 1 } of arity up to the cardinality of the model is called a complete atomic Boolean algebra, or " CABA " . ( In place of this awkward restriction on arity we could allow any arity, leading to a different awkwardness, that the signature would then be larger than any set, that is, a proper class.
19.
These different " equational theories " of types vary widely, two extreme cases being " structural type systems ", in which any two types that describe values with the same structure are equivalent, and " nominative type systems ", in which no two syntactically distinct type expressions denote the same type ( " i . e . ", types must have the same " name " in order to be equal ).
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