# Algebra

##### Objectives and outcomes

Acquiring general and professional knowledge of algebra. Students will acquire important concepts and knowledge of both general algebra and various types of algebraic structures. General algebra – notion of algebraic structure, algebraic law, homomorphism of algebraic structures, congruence, quotient structure, the result of algebraic structures. Special algebraic structures – semigroup, monoid, group, ring, module over a ring, field, network, Boolean algebra. The emphasis is put on basic theorems of group theory.

##### Lectures

Algebraic operations. Algebraic laws. External operation. Sub-operation. Algebraic structure. Substructure. Homomorphisms. Types of homomorphisms. Compositions of homomorphisms. Direct and inverse images. Isomorphism as an equivalence relation on the classes of algebraic structures. Congruences. Quotient structures. Natural homomorphism. Kernel of a homomorphism. Homomorphism decomposition theorem. Direct product of algebras. Groupoids. Semigroups. Regularity of elements in semigroups. Monoids. Invertibility of elements in monoids. Submonoids. Monoid homomorphisms. Groups. Subgroups. Cosets of a subgroup. Lagrange’s theorem. Group homomorphisms. Order of an element. Cyclic groups. Theorems on cyclic groups. Normal subgroups. Correspondence between normal subgroups and congruences. Rings. Examples. Modules over rings. Fields. Examples. A network as a partially ordered set and as an algebraic structure. Complete lattices. Boolean algebra. Trivial and two-element Boolean algebra.