$x$ $y$ $z$ $u$ $v$ $w$ $x_0$ $x_1$ $x_2$ …

• Constant symbols
  • $a$ $b$ $c$ $d$ $e$ $\bot$ $\top$ $\emptyset$ $\infty$ $0$-$9$ $\alpha-\omega$

• Unary operation symbols
  • Prefix: $f$ $g$ $h$ $-$ $\neg$ $\sim$
  • Postfix: $'$ ${}^{-1}$ ${}^{\cup}$

• Binary operation symbols
  • Prefix: $f$ $g$ $h$
  • Infix: $+$ $-$ $*$ $\cdot$ $\times$ $\div$ $/$ $\backslash$ $\circ$ $\oplus$ $\otimes$ $\odot$ $\wedge$ $\vee$ $\to$

• Ternary operation symbols
  • Prefix: $t$

• Quadternary operation symbols
  • Prefix: $q$

• $n$-ary operation symbols
  • Prefix: $\sum_n$ $\prod_n$
  • Mixfix: { } [ ] ( )

• $\omega$-ary operation symbols
  • $\sum_\omega$ $\prod_\omega$

• $\kappa$-ary operation symbols
  • $\sum_\kappa$ $\prod_\kappa$

• $\infty$-ary operation symbols
  • $\sum$ $\prod$ $\bigcap$ $\bigcup$ $\bigwedge$ $\bigvee$

• 0-ary relation symbols
  • Propositions ⊥ ⊤ T F

• Unary relation symbols
  • Prefix: P Q

• Binary relation symbols
  • Prefix: R
  • Infix: = ≠ ≤ < ≥ > ⪯ ≺ ⪰ ≻ ≡ ≅ ≈ ∈ ∉

• Ternary relation symbols

• Quadternary relation symbols

• n-ary relation symbols

• ω-ary relation symbols

• κ-ary relation symbols

• ∞-ary relation symbols

• 0-ary connectives

• Unary connectives
  • ⋄ □

• Binary connectives
  • Infix: and ∧ or ∨ ⇒ ⇔
  • Mixfix: if_then_ &nbsp; while_do_

• Ternary connectives
  • if_then_else_

• Quadternary connectives

• n-ary connectives

• ω-ary connectives

• κ-ary connectives

• ∞-ary connectives

• ∀ ∃

• ( ) [ ] { } 〈 〉

A signature or language is a sequence of operation symbols, relation symbols and connectives.

Prefix symbols always apply first. Infix symbols and postfix symbols can

In a typed (or multisorted) language the input/output slots of operation symbols and relation symbols are associated to a specific type.

In an order-sorted language the types form a partially ordered set that determines the subtyping relation.