Subdirectly irreducible commutative idempotent semirings

Transcription

1 Subdirectly irreducible commutative idempotent semirings Ivan Chajda Helmut Länger Palacký University Olomouc, Olomouc, Czech Republic, Vienna University of Technology, Vienna, Austria, Support of the research of both authors by the Austrian Science Fund (FWF) and the Czech Science Foundation (GAČR), project I 1923-N25, and by AKTION Austria Czech Republic, grant No. 71p3, is gratefully acknowledged. 90th Workshop on General Algebra, University of Novi Sad, June 7, 2015 Chajda, Länger Subdirectly irreducible semirings 1 / 20

2 Contents: 1. Semirings 2. Varieties of semirings 3. Subdirectly irreducible semirings 4. Concluding remarks 5. References Chajda, Länger Subdirectly irreducible semirings 2 / 20

3 1. Semirings Chajda, Länger Subdirectly irreducible semirings 3 / 20

4 Definition of a semiring and examples Definition 1 A semiring is an algebra (R, +,, 0, 1) of type (2, 2, 0, 0) satisfying (R, +, 0) is a commutative monoid. (R,, 1) is a monoid. The operation is distributive with respect to +. x0 = 0x = 0 Example 2 ({0, 1, 2, 3,...}, +,, 0, 1) is a semiring. Every unitary ring is a semiring. Every bounded distributive lattice is a semiring. Chajda, Länger Subdirectly irreducible semirings 4 / 20

5 2. Varieties of semirings Chajda, Länger Subdirectly irreducible semirings 5 / 20

6 Different kinds of semirings Definition 3 A semiring is called Let commutative if is commutative idempotent if is idempotent Boolean if it is commutative and idempotent and additionally satisfies 1 + x + x = 1 trivial if it has only one element S denote the variety of semirings C the variety of commutative idempotent semirings B the variety of Boolean semirings V the subvariety of C determined by xy + x + 1 = x + 1 T the variety of trivial semirings Chajda, Länger Subdirectly irreducible semirings 6 / 20

7 Hasse diagram of semiring varieties We have the following Hasse diagram: S C B B V V B V T Chajda, Länger Subdirectly irreducible semirings 7 / 20

8 3. Subdirectly irreducible semrings Chajda, Länger Subdirectly irreducible semirings 8 / 20

9 Subdirectly irreducible algebras Definition 4 An algebra A with base set A is called subdirectly irreducible (SI) if there exists a smallest (with respect to ) congruence Θ on A with Θ (:= {(x, x) x A}), the so-called monolith of A. The importance of knowing all SI members of a variety is expressed by the following well-known fact: Theorem 5 Every variety is generated by its SI members. Therefore it is interesting to know all SI members of B, C and V. Chajda, Länger Subdirectly irreducible semirings 9 / 20

10 Structure of SI members of C Lemma 6 If (R, +,, 0, 1) C then (R, ) is a semilattice. We consider this semilattice as a meet-semilattice. Let denote the corresponding partial order relation. Then (R,, 0, 1) is a bounded poset. Lemma 7 If R = (R, +,, 0, 1) is an SI member of C then there exists a coatom a of (R, ) such that R = [0, a] {1} {a, 1} 2 is the monolith of R. Corollary 8 If (R, +,, 0, 1) is an SI member of C and R 4 then (R, ) is a chain. Chajda, Länger Subdirectly irreducible semirings 10 / 20

11 Definition of S n and T n Definition 9 For every integer n > 1 put S n := {1,..., n} and let 1 denote the linear ordering on S n given by } { n 1 1 n 1 n even if n is n 1 n 1 1 n odd Moreover, put x + 1 y := max 1 (x, y) { n 1 if x = y = n x + 2 y := x + 1 y otherwise xy := min (x, y) S n := (S n, + 1,, 1, n) T n := (S n, + 2,, 1, n) Chajda, Länger Subdirectly irreducible semirings 11 / 20

12 Definition of S C and T C Definition 10 For every infinite bounded chain C = (C, 2, 0, 1) let S C denote the algebra (S C, +,, (0, 1), (1, 2)) of type (2, 2, 0, 0) defined by S C := C {1, 2}, (x, i) + (y, j) := (x, i)(y, j) := (max 2 (x, y), 1) (y, 2) (x, 2) (min 2 (x, y), 2) (x, i) (x, min(i, j)) (y, j) if x if (i, j) = < = > y (1, 1) (1, 2) (2, 1) (2, 2) ((x, i), (y, j) S C ). Moreover, let T C denote the algebra of type (2, 2, 0, 0) which coincides with S C with the only exception that (1, 2)+(1, 2) := (1, 1) instead of (1, 2) + (1, 2) := (1, 2). Chajda, Länger Subdirectly irreducible semirings 12 / 20

13 Definition of B 1 Definition 11 For any non-trivial Boolean lattice B = (B,,, 0, a) let B 1 denote the semiring (S, +,, 0, 1) where 1 / B, S := B {1} and + and are defined as follows: x + y := xy := x y 1 a x y y x if if x, y 1 (x, y) {(0, 1), (1, 0)} otherwise x, y 1 x = 1 y = 1 Chajda, Länger Subdirectly irreducible semirings 13 / 20

14 SI semirings Theorem 12 (Guzmán 92) Up to isomorphism S 2 and T 2 are all SI members of B. S n, T n, S C, T C and B 1 are SI members of C. S 2, T 3 and B 1 are SI members of V. Remark 13 If n 4 then up to isomorphism S n and T n are the only n-element SI members of C. If C is an n-element chain then S C = S2n and T C = T2n. T 3 = 2 1 S n V if and only if n = 2 T n V if and only if n = 3 S C, T C / V Chajda, Länger Subdirectly irreducible semirings 14 / 20

15 4. Concluding remarks Chajda, Länger Subdirectly irreducible semirings 15 / 20

16 Number of SI semirings Remark 14 Up to isomorphism B has exactly two SI members. Up to isomorphism C has at least two SI members of cardinality n > 1. If m 2 then up to isomorphism C has at least three SI members of cardinality 2 m + 1. If m 0 then up to isomorphism V has at least one SI member of cardinality 2 m + 1. Since for infinite n there exist 2 n pairwise non-isomorphic Boolean algebras of cardinality n, V has at least 2 n SI members of infinite cardinality n. Chajda, Länger Subdirectly irreducible semirings 16 / 20

17 Hasse diagram of semiring varieties revisited Remark 15 All inclusions in the Hasse diagram S C B V B V B V are proper. T Chajda, Länger Subdirectly irreducible semirings 17 / 20

18 5. References Chajda, Länger Subdirectly irreducible semirings 18 / 20

19 References [1] I. Chajda, H. Länger and F. Švrček, Multiplicatively idempotent semirings. Math. Bohemica 140 (2015), [2] J. S. Golan, Semirings and Their Applications. Kluwer, Dordrecht ISBN [3] F. Guzmán, The variety of Boolean semirings. J. Pure Appl. Algebra 78 (1992), [4] W. Kuich and A. Salomaa, Semirings, Automata, Languages. Springer, Berlin ISBN [5] J. D. Monk and R. Bonnet (eds.), Handbook of Boolean Algebras. Vol. 2. North-Holland, Amsterdam ISBN Chajda, Länger Subdirectly irreducible semirings 19 / 20

20 Thank you for your attention! Chajda, Länger Subdirectly irreducible semirings 20 / 20