Reflexive: ∀x x ≤ xAntisymmetric: ∀x ∀y x ≤ y ∧ y ≤ x → x = yPartial: Transitive ∧ Reflexive ∧ Antisymmetric; Linear (or total): Partial ∧ ∀x ∀y x ≤ y ∨ y ≤ xDense: ∀x ∀z x < z → ∃y x < y ∧ y < z ("Between any 2 distinct elements there is another element") There is a smallest element: ∃x ∀y x ≤ yThere is a largest element: ∃x ∀y y ≤ xEvery element has an immediate successor: ∀x ∃y ∀z x < z ↔ y ≤ zThe theory DLO of dense linear orders without endpoints (i.e. no smallest or largest element) is complete, ω-categorical, but not categorical for any uncountable cardinal. There are three other very similar theories: the theory of dense linear orders with a: Smallest but no largest element; Largest but no smallest element; Largest and smallest element. Being well ordered ("any non-empty subset has a minimal element") is not a first-order property; the usual definition involves quantifying over all subsets. Lattices Lattices can be considered either as special sorts of partially ordered sets, with a signature consisting of one binary relation symbol ≤, or as algebraic structures with a signature consisting of two binary operations ∧ and ∨. The two approaches can be related by defining a ≤ b to mean a∧b = a. For two binary operations the axioms for a lattice are: For one relation ≤ the axioms are: Axioms stating ≤ is a partial order, as above. (existence of c = a∧b) (existence of c = a∨b) First order properties include: (distributive lattices) (modular lattices) Heyting algebras can be defined as lattices with certain extra first-order properties. Completeness is not a first order property of lattices. Graphs The signature of graphs has no constants or functions, and one binary relation symbol R, where R(x,y) is read as "there is an edge from x to y". The axioms for the theory of graphs are Symmetric: ∀x ∀y R(x,y)→ R(y,x) Anti-reflexive: ∀x ¬R(x,x) ("no loops") The theory of random graphs has the following extra axioms for each positive integer n: For any two disjoint finite sets of size n, there is a point joined to all points of the first set and to no points of the second set. (For each fixed n, it is easy to write this statement in the language of graphs.) The theory of random