1. **Problem Statement:** Define Group, Semigroup, Monoid, Subgroup, Abelian Group, Cyclic Group, and Lattice along with their properties. 2. **Definitions and Properties:** - **Semigroup:** A set $S$ with an associative binary operation $*$, i.e., for all $a,b,c \in S$, $(a * b) * c = a * (b * c)$. - **Monoid:** A semigroup with an identity element $e$ such that for all $a \in S$, $e * a = a * e = a$. - **Group:** A monoid where every element has an inverse. For each $a \in G$, there exists $a^{-1} \in G$ such that $a * a^{-1} = a^{-1} * a = e$. - **Subgroup:** A subset $H$ of a group $G$ that is itself a group under the operation of $G$. - **Abelian Group:** A group $G$ where the operation is commutative: for all $a,b \in G$, $a * b = b * a$. - **Cyclic Group:** A group generated by a single element $g$, i.e., every element in $G$ can be written as $g^n$ for some integer $n$. - **Lattice:** A partially ordered set $(L, \leq)$ where every pair of elements has a unique least upper bound (join) and greatest lower bound (meet). 3. **Summary:** - Semigroup: associative operation. - Monoid: semigroup + identity. - Group: monoid + inverses. - Subgroup: subset that is a group. - Abelian group: group + commutativity. - Cyclic group: generated by one element. - Lattice: poset with join and meet for every pair.