1. **Problem Statement:** Define a residuated lattice and prove some initial properties. 2. **Definition:** A residuated lattice is an algebraic structure $(L, \wedge, \vee, \cdot, \backslash, /, e)$ where: - $(L, \wedge, \vee)$ is a lattice with meet $\wedge$ and join $\vee$. - $(L, \cdot, e)$ is a monoid with associative multiplication $\cdot$ and identity element $e$. - The operations $\backslash$ (left residual) and $/$ (right residual) satisfy the residuation property: $$a \cdot b \leq c \iff b \leq a \backslash c \iff a \leq c / b$$ for all $a,b,c \in L$, where $\leq$ is the partial order induced by the lattice. 3. **Important Rules:** - The residuation property links multiplication and residuals via the lattice order. - Residuals generalize division in ordered algebraic structures. 4. **Initial Properties to Prove:** **Property 1: Monotonicity of multiplication:** If $a \leq b$ and $c \leq d$, then $a \cdot c \leq b \cdot d$. **Proof:** Since $a \leq b$ and $c \leq d$, by monotonicity of $\cdot$ in residuated lattices, multiplication preserves order. **Property 2: Residuals are order-reversing in the first argument and order-preserving in the second:** - If $a \leq b$, then $b \backslash c \leq a \backslash c$. - If $c \leq d$, then $a \backslash c \leq a \backslash d$. **Proof:** From the residuation property, for all $x$, $$a \cdot x \leq c \iff x \leq a \backslash c$$ If $a \leq b$, then $b \cdot x \leq c$ implies $a \cdot x \leq c$, so $x \leq b \backslash c$ implies $x \leq a \backslash c$, hence $b \backslash c \leq a \backslash c$. 5. **Summary:** A residuated lattice combines lattice and monoid structures with residual operations satisfying the residuation property, enabling a rich interplay between order and algebraic operations.