The paper discusses the functional semantics of algebraic theories, introduced by F. William Lawvere. It explores how algebraic structures can be studied using category theory, particularly through the use of adjoint functors. The author shows that many algebraic constructions, such as free algebras and monoid rings, can be viewed as adjoints to algebraic functors. By formalizing "semantics" as a functor and showing it has an adjoint, the author provides a new characterization of equational classes of algebras and a tool for analyzing nonalgebraic categories. An algebraic theory is defined as a category with objects being natural numbers and each object n being the product of 1 with itself n times. An n-ary operation is a map from n to 1 in the category. Algebraic theories and mappings between them form a category. Each algebraic theory determines a category of algebras, and mappings between theories determine functors between these categories. The paper proves that every algebraic functor has an adjoint, and that algebraic semantics has an adjoint, which is called algebraic structure. This adjoint is naturally equivalent to the identity functor on the category of algebraic theories. The paper also discusses the characterization of algebraic categories and provides examples, such as the category of rings and monoids. It shows that any category with finite limits and an abstractly finite regular projective generator is equivalent to an algebraic category. The paper also discusses the relationship between algebraic categories and abelian categories, and the nature of rings implied by these results. The paper concludes with a corollary that an abelian category is algebraic if and only if it is the category of all modules over some associative ring with unity. The paper also references related work by other authors and discusses the implications of these results for the study of algebraic structures.The paper discusses the functional semantics of algebraic theories, introduced by F. William Lawvere. It explores how algebraic structures can be studied using category theory, particularly through the use of adjoint functors. The author shows that many algebraic constructions, such as free algebras and monoid rings, can be viewed as adjoints to algebraic functors. By formalizing "semantics" as a functor and showing it has an adjoint, the author provides a new characterization of equational classes of algebras and a tool for analyzing nonalgebraic categories. An algebraic theory is defined as a category with objects being natural numbers and each object n being the product of 1 with itself n times. An n-ary operation is a map from n to 1 in the category. Algebraic theories and mappings between them form a category. Each algebraic theory determines a category of algebras, and mappings between theories determine functors between these categories. The paper proves that every algebraic functor has an adjoint, and that algebraic semantics has an adjoint, which is called algebraic structure. This adjoint is naturally equivalent to the identity functor on the category of algebraic theories. The paper also discusses the characterization of algebraic categories and provides examples, such as the category of rings and monoids. It shows that any category with finite limits and an abstractly finite regular projective generator is equivalent to an algebraic category. The paper also discusses the relationship between algebraic categories and abelian categories, and the nature of rings implied by these results. The paper concludes with a corollary that an abelian category is algebraic if and only if it is the category of all modules over some associative ring with unity. The paper also references related work by other authors and discusses the implications of these results for the study of algebraic structures.