Semigroup of Tropical Matrices Munazza Naz University of York NBSAN, July 13, 2017 Joint work with Victoria Gould and Marianne Johnson Munazza Naz (University of York) Semigroup of Tropical Matrices NBSAN, July 13, 2017 1 / 27 Contents Exact Semirings and Information Algebras Tropical Semield Tropical Algebra Upper Triangular Tropical Matrices Un(FT) Green's Equivalence Relations on Un(FT) Idempotent and Regular Matrices in Un(FT) ∗− and ∼ − Extensions of Green's Relations on Un(FT) Munazza Naz (University of York) Semigroup of Tropical Matrices NBSAN, July 13, 2017 2 / 27 Extension of Green's Relations On a semigroup S, the relation ≤R∗ is dened by the rule that for a, b 2 S, a ≤R∗ b if and only if xb = yb ) xa = ya and xb = b ) xa = a for all x, y 2 S. Clearly ≤R∗ is a quasi-order on S. ∗ The equivalence relation associated with ≤R∗ is denoted by R and it is a left congruence on S. It follows, a R b implies that a R∗ b and, e R f if and only if e R∗ f for all e, f 2 E (S). The relation L∗ is dened dually and the relations H∗ and D∗ by putting H∗ = R∗\L∗; D∗ = R∗ _L∗. Munazza Naz (University of York) Semigroup of Tropical Matrices NBSAN, July 13, 2017 3 / 27 Extension of Green's Relations For any a 2 S, the set of left identities for a in E ⊆ E (S) is denoted by E a, that is, E a = fe 2 E : ea = ag. The relation ≤ on S is dened by the rule that for a, b 2 S, a ≤ b if Re Re and only if for all e 2 E eb = b ) ea = a. The equivalence relation associated with is denoted by . We have ≤R ReE following result: e Lemma For any a 2 S and e 2 E, a Re e if and only if ea = a and for all f 2 E, if fa = a then fe = e. Munazza Naz (University of York) Semigroup of Tropical Matrices NBSAN, July 13, 2017 4 / 27 Extension of Green's Relations We say that S is left E-abundant if every R∗-class contains an idempotent of E; Weakly left E-abundant if every Re-class contains an idempotent of E; abundant if each R∗-class and each L∗-class of S contains an idempotent; A semigroup S is weakly abundant (Fountain Semigroup!) if each Re-class and each Le-class of S contains an idempotent. For basic facts about the relations and abundant and weakly abundant semigroups we refer to: 1 J. Fountain, A class of right PP monoids, Quart. J. Math. Oxford (2) 28 (1977), pp 285-300. 2 J. Fountain, Adequate semigroups, Proc. Edinb. Math. Soc. (2) 22 (1979), pp 113-125. Munazza Naz (University of York) Semigroup of Tropical Matrices NBSAN, July 13, 2017 5 / 27 Semirings A hemiring [resp. semiring] is a nonempty set S on which operations of addition and multiplication have been dened such that the following conditions are satised: 1 (S, +) is a commutative monoid with identity element 0; 2 (S, ·) is a semigroup [resp. monoid with identity element 1S ]; 3 Multiplication distributes over addition from either side; 4 0r = 0 = r0 for all r 2 S. As a rule, we will write 1 instead of 1S when there is no likelihood of confusion. Note that if 1 = 0 then r = r1 = r0 = 0 for each element r of S and so S = f0g. In order to avoid this trivial case, we will assume that all semirings under consideration are nontrivial, i.e. 5.1 6= 0. Munazza Naz (University of York) Semigroup of Tropical Matrices NBSAN, July 13, 2017 6 / 27 Information Algebras A semiring S is said to be zerosumfree if and only if r + r 0 = 0 implies that r = r 0 = 0 . Every additively idempotent semiring is zerosum free. If S is a zerosumfree semiring then S 0 = f0g [ fr 2 S : rb 6= 0 for all 0 6= b 2 Sg is a subsemiring of S. A nonzero element a of a semiring S is a left(right) zero divisorif and only if there exists a nonzero element b of S satisfying ab = 0(ba = 0). It is a zero divisor if and only if it is a left and a right zero divisor. A semiring S having no zero divisors is entire. If S is an entire, zerosumfree semiring thenS ∗ = S − f0g is a subsemiring of S. Entire zerosumfree semirings are called information algebras. Munazza Naz (University of York) Semigroup of Tropical Matrices NBSAN, July 13, 2017 7 / 27 Semigroup of Matrices over Semirings Theorem A division semiring S is either zerosumfree or is a division ring. Let Mn(S) denote the set of all n × n matrices over a semiring S and the induced operation of multiplication be dened as: n (AB)ij = ∑ (Aik Bkj ) k=1 for all A, B 2 Mn(S). Theorem Let S be a semiring and n 2 N. Then: 1 Mn(S) is a semigroup under induced operation of multiplication; 2 Set of upper triangular matrices where entries below the diagonal are zero, denoted Un(S) is a subsemigroup of Mn(S). Munazza Naz (University of York) Semigroup of Tropical Matrices NBSAN, July 13, 2017 8 / 27 Semigroup of Matrices over Semirings Let G be a reexive, transitive relation on [n] := f1, ..., ng, and dene G(S) = fA 2 Mn(S) : Aij 6= 0 , (i, j) 2 Gg. Theorem Let S be an information algebra and Mn(S) be the set of n × n matrices over S. Then the set G(S) is a subsemigroup of Mn(S). Denition A semiring S is said to be exact (or FP-injective) if for all A 2 Sm×n: (E1) If x 2 S1×nnRow(A), then there exist u,v 2 Sn×1 such that Au = Av, but xu 6= xv. (E2) If x 2 Sm×1nCol(A), then there exist u,v 2 S1×m such that uA = vA, but ux 6= vx. Munazza Naz (University of York) Semigroup of Tropical Matrices NBSAN, July 13, 2017 9 / 27 Example of Exact Semirings The class of exact semirings is known to contain: all elds; all proper quotients of principal ideal domains; all matrix rings and nite group rings over the above; the Boolean semiring B, the tropical semiring T, and some generalisations of these. For more information on semirings and exact semirings we refer to: 1 J. S. Golan, Semirings and their Applications, Springer Science Business Media, 1999. 2 D. Wilding, M. Johnson, M. Kambites, Exact rings and semirings, J. Algebra 388 (2013), 324-337. Munazza Naz (University of York) Semigroup of Tropical Matrices NBSAN, July 13, 2017 10 / 27 Green's Relations on Semigroup of Matrices over Semirings Two matrices A, B 2 Mn(S) are L-related, written A L B, if there are M, P 2 Mn(S) with MA = B and PB = A. OR A L B if and only if Row(A) = Row(B). Similarly A, B 2 Mn(S) are R-related, written A R B, if there are N, Q 2 Mn(S) with AN = B and BQ = A. Again, we notice that A R B if and only if Col(A) = Col(B). Theorem Let S be an exact semiring. Then R = R∗ and L = L∗ in the semigroup Mn(S). Thus Mn(S) is abundant if and only if it is regular. Munazza Naz (University of York) Semigroup of Tropical Matrices NBSAN, July 13, 2017 11 / 27 Tropical Semield We focus here on S = FT. The nitary tropical (or max-plus) semield FT has elements from R with binary operations dened as: x ⊕ y = max(x, y); and x ⊗ y = x + y. We see that (FT, ⊕, ⊗) is an idempotent semield. Its generalisations are (T, ⊕, ⊗) and (T, ⊕, ⊗), where T = R [ f−¥g and T = R [ f−¥, ¥g. Munazza Naz (University of York) Semigroup of Tropical Matrices NBSAN, July 13, 2017 12 / 27 Conventions Here we have some conventions: a ⊕ −¥ = a = −¥ ⊕ a, a ⊕ ¥ = ¥ = ¥ ⊕ a, ¥ ⊕ −¥ = ¥ = −¥ ⊕ ¥, a ⊗ −¥ = −¥ = −¥ ⊗ a, a ⊗ ¥ = ¥ = ¥ ⊗ a, ¥ ⊗ −¥ = −¥ = −¥ ⊗ ¥, for all a 2 FT. Munazza Naz (University of York) Semigroup of Tropical Matrices NBSAN, July 13, 2017 13 / 27 Tropical Matrices and Vectors Let Mn(S) denotes the set of all n × n matrices over S 2 fFT, T, T, g, with multiplication ⊗ dened as: n M (A ⊗ B)ij = (Aik ⊗ Bkj ) k=1 for all A, B 2 Mn(S). Then (Mn(S), ⊗) is a semigroup. Tropical Ane n-Space n Let S denote the set of all real n-tuples ~v = (v1, ..., vn) with obvious operations of addition and scalar multiplication: (~v ⊕ w~ )i = vi ⊕ wi , (l ⊗~v)i = l ⊗ vi . Munazza Naz (University of York) Semigroup of Tropical Matrices NBSAN, July 13, 2017 14 / 27 Semigroup of Tropical Matrices For A 2 Mn(S) The row space R(A) ⊆ Sn is dened as the tropical submodule of Sn generated by the rows of A. And similarly, The column space C (A) ⊆ Sn is dened as the tropical submodule Sn generated by the columns of A. Example Let 2 2 4 5 3 2 7 6 7 3 A = 4 5 3 7 5 and B = 4 −¥ 4 9 5 . 9 ¥ 1 6 3 3 Then, 2 11 8 13 3 AB = 4 13 11 12 5 . 16 ¥ ¥ Munazza Naz (University of York) Semigroup of Tropical Matrices NBSAN, July 13, 2017 15 / 27 Green's Equivalence Relations Known Characterisations of Green's Relations. Theorem Let A, B 2 Mn(S) for S 2 fFT, T, T¯ g.