Why a subobject is not enough
For groups you quotient by a normal subgroup; for rings, by an ideal. These look like different machinery, but both are doing the same thing: specifying *which elements get glued together*. The honest object describing a quotient is an equivalence relation θ on A that is compatible with the operations. Such a θ is a congruence, and it works for every signature — even ones, like lattices or semigroups, where no kernel-as-subset description exists.
Concretely, θ ⊆ A × A is a congruence if it is an equivalence relation and is *compatible*: for every n-ary operation ω, whenever aᵢ θ bᵢ for all i, then ω(a₁,…,aₙ) θ ω(b₁,…,bₙ). Compatibility is exactly the condition you need to define operations on equivalence classes without ambiguity. Then the quotient algebra A/θ has underlying set the classes [a]_θ, with ω([a₁],…,[aₙ]) := [ω(a₁,…,aₙ)], and the projection a ↦ [a] is a surjective homomorphism.
The isomorphism theorems, once
Every homomorphism f: A → B has a kernel congruence ker f := { (a, a′) : f(a) = f(a′) }. This is always a congruence (check compatibility directly from the homomorphism property). The first isomorphism theorem then reads, with no per-structure changes: A / ker f ≅ image(f), via [a] ↦ f(a). The Volume I theorems for quotient rings and for groups are the same statement read in two signatures.
Lattice example where congruences, not subobjects, run the show.
Let L be the chain 0 < a < b < 1 (a 4-element lattice, join = max, meet = min).
Define theta by collapsing a and b together, keeping 0 and 1 apart:
classes: {0}, {a, b}, {1}
Check compatibility for join (max) and meet (min):
a theta b. Take join with 1: a v 1 = 1, b v 1 = 1. 1 theta 1. OK
a theta b. Take meet with 0: a ^ 0 = 0, b ^ 0 = 0. 0 theta 0. OK
a theta b. join with a: a v a = a, b v a = b. a theta b. OK
Every operation respects theta, so theta is a congruence.
Quotient L/theta is the 3-element chain {0} < {a,b} < {1}.
The projection L -> L/theta is a surjective lattice homomorphism, and
there is NO subset of L whose 'class of bottom' alone records this gluing
-- you genuinely need the relation theta itself.The congruence lattice
The set Con(A) of all congruences on A, ordered by inclusion, is itself a lattice — in fact a complete lattice: arbitrary intersections of congruences are congruences (so meets exist), and the join of a family is the least congruence containing their union. The bottom is the diagonal Δ = { (a,a) } (where A/Δ ≅ A) and the top is A × A (where A/(A×A) is the one-element algebra). The structure of Con(A) is a deep invariant; for instance A is simple in the universal-algebra sense precisely when Con(A) has exactly two elements.