Equivalence relation:
An equivalence relation is a relation that is reflexive, symmetric, and transitive. Let's check if the ordering relation a|bSNxN over natural numbers N satisfies these properties:

1. Reflexivity: For any natural number a, a|a if there exists c ∈ N such that a*c = a. Since a*1 = a, we can see that a|a. Therefore, the relation is reflexive.

2. Symmetry: If a|b, then there exists c ∈ N such that a*c = b. Taking the reciprocal, we have b*(1/c) = a. As 1/c is also a natural number, we can see that b|a. Hence, the relation is symmetric.

3. Transitivity: If a|b and b|c, then there exist c1, c2 ∈ N such that a*c1 = b and b*c2 = c. Multiplying these equations, we get a*(c1*c2) = c. As the product of two natural numbers is also a natural number, we can see that a|c. Therefore, the relation is transitive.

Since the ordering relation a|bSNxN satisfies all three properties (reflexivity, symmetry, and transitivity), it is an equivalence relation.

Total order:
A total order is a relation that is reflexive, antisymmetric, and transitive. Let's check if the ordering relation a|bSNxN over natural numbers N satisfies these properties:

1. Reflexivity: We have already established that the relation is reflexive.

2. Antisymmetry: If a|b and b|a, then there exist c1, c2 ∈ N such that a*c1 = b and b*c2 = a. Multiplying these equations, we get a*(c1*c2) = a. Since a is a natural number, c1*c2 must be equal to 1. This implies that both c1 and c2 are 1. Therefore, a = b. Hence, the relation is antisymmetric.

3. Transitivity: We have already established that the relation is transitive.

Since the ordering relation a|bSNxN satisfies all three properties (reflexivity, antisymmetry, and transitivity), it is a total order.

Lattice:
A lattice is a partially ordered set in which every pair of elements has a unique supremum (least upper bound) and infimum (greatest lower bound). Let's check if the ordering relation a|bSNxN over natural numbers N forms a lattice:

- Supremum: Given any two natural numbers a and b, their supremum (least upper bound) under the ordering relation a|bSNxN can be determined by finding their least common multiple (LCM). The LCM of a and b is the smallest natural number that is divisible by both a and b. Therefore, the supremum of a and b exists and is unique.

- Infimum: Given any two natural numbers a and b, their infimum (greatest lower bound) under the ordering relation a|bSNxN can be determined by finding their greatest common divisor (GCD). The GCD of a and b is the largest natural number that divides both a and b. Therefore, the infimum of a and b exists and is