Lattice Algebra: Theory and Applications Prof. Gerhard Ritter CISE Department, University of Florida Lattice Theory & Applications – p. 1/87 Overview Part I: Theory • Pertinent algebraic structures • Lattice algebra with focus on ℓ-vector Spaces • Concluding remarks and questions Part II: Applications • LNNs • Matrix based LAMs • Dendritic LAMs • Concluding remarks and questions Lattice Theory & Applications – p. 2/87 History • Lattice theory in image processing and AI • Image algebra, mathematical morphology, and HPC A pertinent question: Why is (−1) · (−1) = 1? Lattice Theory & Applications – p. 3/87 Algebraic Structures Some basic backgound Let G be a set with binary operation ◦. Then 1. (G, ◦) is a groupoid 2. if x ◦ (y ◦ z) = (x ◦ y) ◦ z, then (G, ◦) is a semigroup 3. if G is a semigroup and and G has an identity element, then G is a monoid 4. if G is a monoid and every element of G has an inverse, then G is a group 5. if G is a group and x ◦ y = y ◦ x ∀ x, y ∈ G, then G is an abelian group. Lattice Theory & Applications – p. 4/87 Algebraic Structures Why are groups important? Theorem. If (X, ·) is a group and a, b ∈ X, then the linear equations a · x = b and y · a = b have unique solutions in X. Remark: Note that the solutions x = a−1 · b and y = b · a−1 need not be the same unless X is abelian. Lattice Theory & Applications – p. 5/87 Algebraic Structures Sets with Multiple Operations Suppose that X is a set with two binary operations ⋆ and ◦. The operation ◦ is said to be left distributive with respect to ⋆ if x ◦ (y⋆z) = (x ◦ y) ⋆ (x ◦ z) ∀ x, y, z ∈ X (1) and right distributive if (y⋆z) ◦ x = (y ◦ x) ⋆ (z ◦ x) ∀ x, y, z ∈ X. (2) Division on R+ is not left distributive over addition; (y + z)/x = (y/x) + (z/x) but x/(y + z) =6 (x/y) + (x/z). When both equations hold, we simply say that ◦ is distributive with respect to ⋆. Lattice Theory & Applications – p. 6/87 Algebraic Structures Definition: A ring (R, +, ·) is a set R together with two binary operations + and · of addition and multiplication, respectively, defined on R such that the following axioms are satisfied: 1. (R, +) is an abelian group. 2. (R, ·) is a semigroup. 3. ∀ a, b, c ∈ R, a · (b + c) = (a · b) + (a · c) and (a + b) · c = (a · c) + (b · c). If axiom 1 in this definition is weakened to (R, +) is a commutative semigroup, then R is called a semiring. Lattice Theory & Applications – p. 7/87 Algebraic Structures If (R, +, ·) is a ring, we let 0 denote the additive identity and 1 the multiplicative identity (if it exists). If R satisfies the property • For every nonzero a ∈ R there is an element in R, denoted by a−1, such that a · a−1 = a−1 · a = 1 (i.e. (R \{0}, ·) is a group), then R is called division ring. A commutative division ring is called a field You should now be able to prove that (−1) · (−1) = 1. Lattice Theory & Applications – p. 8/87 Partially Ordered Sets Definition: A relation 4 on a set X is called a partial order on X if and only if for every x, y, z ∈ X the following three conditions are satisfied: 1. x 4 x (reflexive) 2. x 4 y and y 4 x ⇒ x = y (antisymmetric) 3. x 4 y and y 4 z ⇒ x 4 z (transitive) The inverse relation of 4, denoted by <, is also a partial order on X. Definition: The dual of a partially ordered set X is that partially ordered set X∗ defined by the inverse partial order relation on the same elements. Since (X∗)∗ = X, this terminology is legitimate. Lattice Theory & Applications – p. 9/87 Lattices Definition: A lattice is a partially ordered set L such that for any two elements x, y ∈ L, glb{x, y} and lub{x, y} exist. If L is a lattice, then we define x ∧ y = glb{x, y} and x ∨ y = lub{x, y}. • A sublattice of a lattice L is a subset X of L such that for each pair x, y ∈ X, we have that x ∧ y ∈ X and x ∨ y ∈ X. • A lattice L is said to be complete if and only if for each of its subsets X, infX and supX exist. We define the symbols X = infX and X = supX. V W Lattice Theory & Applications – p. 10/87 sℓ-Semigroups and ℓ-Groups Suppose (R, ◦) is a semigroup or group and R is a lattice (R, ∨, ∧) or semilattice (R, ∨). Definition: A group translation ψ is a function ψ : R → R of form ψ(x)= a ◦ x ◦ b, where a, b are constants. The translation ψ is said to be isotone if and only if x 4 y ⇒ ψ(x) 4 ψ(y) Note that a group translation is a unary operation. Lattice Theory & Applications – p. 11/87 sℓ-Semigroups and ℓ-Groups Definition: A ℓ-group (ℓ-semigroup) is of form (R, ∨, ∧, +), where (R, +) is a group (semigroup) and (R, ∨, ∧) is a lattice, and every group translation is isotone. If R is just a semilattice - i.e., (R, ∨) or (R, ∧) - in the definition, then (R, ∨, +) (or (R, ∧, +)) an sℓ−group if (R, +) is a group and an sℓ-semigroup if (R, +) is a semigroup. Lattice Theory & Applications – p. 12/87 sℓ-Vector Spaces and ℓ-Vector Spaces Definition: A sℓ−vector space V over the sℓ-group (or sℓ-monoid) (R, ∨, +), denoted by V(R),isa semilattice (V, ∨) together with an operation called scalar addition of each element of V by an element of R on the left, such that ∀ α, β ∈ R and v, w ∈ V, the following conditions are satisfied: 1. α + v ∈ V 2. α + (β + v) = (α + β)+ v 3. (α ∨ β)+ v = (α + v) ∨ (β + v) 4. α + (v ∨ w) = (α + v) ∨ (α + w) 5. 0+ v = v Lattice Theory & Applications – p. 13/87 sℓ-Vector Spaces and ℓ-Vector Spaces The sℓ-vector space is also called a max vector space, denoted by ∨-vector space. Using the duals (R, ∧, +) and (V, ∨), and replacing conditions (3.) and (4.) by 3’. (α ∧ β)+ v = (α + v) ∧ (β + v) 4’. α + (v ∧ w) = (α + v) ∧ (α + w), we obtain the min vector space denoted by ∧-vector space. Note also that replacing ∨ (or ∧) by + and + by ·, we obtain the usual axioms defining a vector space. Lattice Theory & Applications – p. 14/87 sℓ-Vector Spaces and ℓ-Vector Spaces Definition: If we replace the semilattice V by a lattice (V, ∨, ∧), the sℓ-group (or sℓ-semigroup) R by an ℓ-group (or ℓ-semigroup) (R, ∨, ∧, +), and conditions 1 through 5 and 3’ and 4’ are all satisfied, then V(R) is called an ℓ-vector space. Remark. The lattice vector space definitions given above are drastically different from vector lattices as postulated by Birkhoff and others! A vector lattice is simply a partially ordered real vector space satisfying the isotone property. Lattice Theory & Applications – p. 15/87 Lattice Algebra and Linear Algebra The theory of ℓ-groups, sℓ-groups, sℓ-semigroups, ℓ-vector spaces, etc. provides an extremely rich setting in which many concepts from linear algebra and abstract algebra can be transferred to the lattice domain via analogies. ℓ-vector spaces are a good example of such an analogy. The next slides will present further examples of such analogies. Lattice Theory & Applications – p. 16/87 Lattice Algebra and Linear Algebra Ring: (R, +, ·) • a · 0 = 0 · a = 0 • a +0=0+ a = a • a · 1 = 1 · a = a • a · (b + c) = (a · b) + (a · c) Semi-Ring or sℓ-Group: (R−∞, ∨, +) • a + (−∞) = (−∞)+ a = −∞ • a ∨ (−∞) = (−∞) ∨ a = a • a +0=0+ a = a • a + (b ∨ c) = (a + b) ∨ (a + c) Lattice Theory & Applications – p. 17/87 Lattice Algebra and Linear Algebra ∗ ∗ ∗ • Since (R−∞, ∨, +) = (R∞, ∧, + ), (R∞, ∧, + ) is also an sℓ−semigroup (with +∗ =+) isomorphic to (R−∞, ∨, +) ∗ • Defining a + b = a + b ∀ a, b ∈ R−∞ and −∞ + ∞ = ∞ + −∞ = −∞ −∞ +∗ ∞ = ∞ +∗ −∞ = ∞, we can combine (R−∞, ∨, +) and (R∞, ∧, +) into one ∗ well defined algebraic structure (R±∞, ∨, ∧, +, + ). Lattice Theory & Applications – p. 18/87 Lattice Algebra and Linear Algebra • The structure (R, ∨, ∧, +) is an ℓ-group. • The structures (R−∞, ∨, ∧, +) and (R∞, ∨, ∧, +) are ℓ-semigroups. • The structure (R±∞, ∨, ∧) is a bounded distributive lattice. ∗ • The structure (R±∞, ∨, ∧, +, + ) is called a bounded lattice ordered group or blog, since the underlying structure (R, +) is a group. Lattice Theory & Applications – p. 19/87 Matrix Addition and Multiplication Suppose A = (aij)m×n and B = (bij)m×n with entries in R±∞. Then • C = A ∨ B is defined by setting cij = aij ∨ bij, and • C = A ∧ B is defined by setting cij = aij ∧ bij. If A = (aij)m×p and B = (bij)p×n, then • C = A ∨ B is defined by setting p cij = k=1(aik + bkj), and • C = AW∧ B is defined by setting p ∗ cij = k=1(aik + bkj). • ∨ andV ∧ are called the max and min products, respectively. Lattice Theory & Applications – p. 20/87 Zero and Identity Matrices For the semiring (Mn×n(R−∞), ∨, ∨ ), the null matrix is −∞ · ··−∞ · −∞·· · Φ= · ···· · ···· −∞ · ··−∞ Lattice Theory & Applications – p. 21/87 Zero and Identity Matrices For the semiring (Mn×n(R−∞), ∨, ∨ ), the identity matrix is 0 −∞· · −∞ −∞ 0 ·· · I = · ··· · · · · · −∞ −∞ · ·−∞ 0 Lattice Theory & Applications – p. 22/87 Matrix Properties We have ∀ A, B, C ∈ Mn×n(R−∞) A ∨ (B ∨ C) = (A ∨ B) ∨ (A ∨ C) I ∨ A = A ∨ I = A A ∨ Φ = Φ ∨ A = A A ∨ Φ = Φ ∨ A = Φ Analogous laws hold for the semiring (Mn×n(R∞), ∧, ∧ ), Lattice Theory & Applications – p. 23/87 Conjugation If r ∈ R±∞, then the additive conjugate of r is the unique element r∗ defined by −r if r ∈ R r∗ = −∞ if r = ∞.