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In universal algebra and mathematical logic, a term algebra is a freely generated algebraic structure over a given
signature.[1][2] For example, in a signature consisting of a single binary operation, the term algebra over a set X of
variables is exactly the free magma generated by X. Other synonyms for the notion include absolutely free algebra,
anarchic algebra.[3]
From a category theory perspective, a term algebra is the initial object for the category of all algebras of the same
signature, and this object, unique up to isomorphism is called an initial algebra; it generates by homomorphic
projection all algebras in the category.[4][5]

A similar notion is that of a Herbrand universe in logic, usually used under this name in logic programming,[6]
which is (absolutely freely) defined starting from the set of constants and function symbols in a set of clauses. That
is, the Herbrand universe consists of all ground terms: terms that have no variables in them.
An atomic formula or atom is commonly defined as a predicate applied to a tuple of terms; a ground atom is then a
predicate in which only ground terms appear. The Herbrand base is the set of all ground atoms that can be formed
from predicate symbols in the original set of clauses and terms in its Herbrand universe. [7][8]
These two concepts are named after Jacques Herbrand.
Term algebras also play a role in the semantics of abstract data types, where an abstract data type declaration
provides the signature of a multi-sorted algebraic structure and the term algebra is a concrete model of the abstract
declaration.
Equality (mathematics) between objects means they are the same in all regards except expression (e.g. 2 +
3 = 5).
Logical equality describes two logical values being the same in Boolean Algebra.
In mathematics, an equation is a formula of the form A = B, where A and B are expressions that may
contain one or several variables called unknowns, and "=" denotes the equality binary relation. Although
written in the form of proposition, an equation is not a statement that is either true or false, but a problem
consisting of finding the values, called solutions, that, when substituted for the unknowns, yield equal
values of the expressions A and B. For example, 2 is the unique solution of the equation x + 2 = 4, in which
the unknown is x.[1] Historically, equations arose from the mathematical discipline of algebra, but later
become ubiquitous. "Equations" should not be confused with "identities", which are presented with the
same notation but have a different meaning: for example 2 + 2 = 4 and x + y = y + x are identities (which
implies they are necessarily true) in arithmetic, and do not constitute a values-finding problem, even when
variables are present as in the latter example.
The term "equation" may also refer to a relation between some variables that is presented as the equality
of some expressions written in terms of those variables' values. For example the equation of the unit circle
is x2 + y2 = 1, which means that a point belongs to the circle if and only if its coordinates are related by this
equation. Most physical laws are expressed by equations. One of the most famous ones is Einstein's
equation E = mc2.
The = symbol was invented by Robert Recorde (1510–1558), who considered that nothing could be more
equal than parallel straight lines with the same length

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