IAENG International Journal of Applied Mathematics, 51:3, IJAM_51_3_05 ______________________________________________________________________________________ The Structure and Characterizaions of Normal Clifford Semirings Jiao Han, Gang Li • Abstract—In this paper, we define normal Clifford semirings, (Rα; +); α 2 D, since E (S) is a normal band, the (S; +) which are generalizations of rectangular Clifford semirings. is a normal orthogroup in which each maximal subgroup is We also give the necessary and sufficient conditions for a + semiring to be normal Clifford semiring and the spined product abelian. So S= D is the distributive lattice D. It is clear that decomposition of normal Clifford semirings. We also discuss a + special case of this kind of semirings, that is strong distributive s D t () V +(s) + s = t + V +(t) lattices of rectangular rings. () (V +(s) + s) \ (t + V +(t)) 6= ;: Index Terms—rectangular rings, normal Clifford semirings, + Distributive lattice congruence, normal band. Due to D is the distributive lattice congruence on semiring + + + S, we get s D s2, st D ts, s(s + t) D s. Foy any c 2 sV +(s), I. INTRODUCTION there exists x 2 V +(s) such that c = sx, from the law of A semiring S = (S; +; •) is an algebra with two binary distribution, we have operation ”+”, ”•” such that the additive reduct (S:+) and s2 + sx + s2 = s(s + x + s) = ss = s2; the multiplicative reduct (S; •) are semigroups connected sx + s2 + sx = s(x + s + x) = sx; by ring-like distributive laws. A semiring S = (S; +; •) is called idempotent semiring, if (8s 2 S)s + s = s = s • s. then sV +(s) ⊆ V +(s2). Hence from [2] and [5] discussed left Clifford semirings and rectangular + 2 + 2 2 + + Clifford semirings respectively. In this paper we will V (s) + s = s + V (s ) ⊇ s + sV (s) = s(s + V (s)); generalize the rectangular Clifford semirings to the normal we can see Clifford semirings. A semiring S is called a normal Clifford semiring if S is the distributive lattices of rectangular band V +(s) + s ⊇ s(s + V +(s)) (8s 2 S): (II.1) semirings and the set of all additive idempotents of S is a normal band. Some structure and characterizations of Also, by normal Clifford semirings and a special case of this kind of V +(st) + st = ts + V +(ts) ⊇ ts + V +(t)s = (t + V +(t))s; semirings will be introduced by us. we have V +(st) + st ⊇ (t + V +(t))s (8s; t 2 S): (II.2) II. CHARACTERIZATIONS AND STRUCTURE Definition II.1. A semiring S is called a normal Clifford From semiring if S is a distributive lattice of a rectangular ring V +(s)+s = s+st+V +(s+st) ⊇ (s+st)+V +(st)+V +(s); and E+(S) is a normal band. we have Similarly, we can get the definition of left normal Clifford semirings. V +(s) + s ⊇ s + st + V +(st) + V +(s)(8s; t 2 S): (II.3) Theorem II.1. A semiring S is a normal Clifford semiring On the other hand, if the additive reduct (S; +) of if and only if the additive reduct (S; +) of S is a normal semiring S is a normal orthogroup in which each maximal + orthogroup in which each maximal subgroup is abelian, subgroup is abelian, then (S; +) is a semilattices S= of + · D E (S) ⊆ E (S) and S satisfies the following conditions. rectangular commutative-groups (S ; +). From the left and + + α (1) 8s 2 S; V (s) + s ⊇ s(s + V (s)); right distributive laws of multiplication over addition, we (2) 8s; t 2 S; V +(st) + st ⊇ (t + V +(t))s; + (S; •) (3) 8s; t 2 S; V +(s) + s ⊇ s + st + V +(st)) + V +(s). obtain that D is a congruence on . If (1) holds, through s2 + V +(s2) ⊇ s2 + sV +(s) = s(s + V +(s)) Proof: Necessity, if S is a normal Clifford semiring, we have then S is a distributive lattice D of rectangular rings Sα; α 2 2 + 2 + D, so E+(S) ⊆ E•(S) and the additive reduct (S; +) of S (s + V (s )) \ (V (s) + s) 6= ;: is an upper semilattice D of rectangular commutative groups + in other words for all s 2 S,s D s2. If (2) holds, then, by Jiao Han is a Master candidate of the school of Mathematics and + + + Statistics, Shandong Normal University, Jinan, 250014, P. R. China (e-mail: ts + V (ts) ⊇ ts + V (t)s = (t + V (t))s; [email protected]). Gang Li is a Professor of the school of Mathematics and Statistics, we have Shandong Normal University, Jinan, 250014, P. R. China(corresponding author to provide e-mail: [email protected]). (ts + V +(ts)) \ (V +(st) + st) 6= ;; Volume 51, Issue 3: September 2021 IAENG International Journal of Applied Mathematics, 51:3, IJAM_51_3_05 ______________________________________________________________________________________ + in other words, for s; t 2 S,st D ts. If (3) holds, then, by Tα; (j; y; µ) 2 Lβ × Tβ × Rβ, then we have s + st + V +(s + st) ⊇ s + st + V +(ts) + V +(s); (i; x; λ) + (j; y; µ) = (i + j; x + y; λ + µ) we have where i + j(x + y; λ + µ) is the sum of i and j (x and y, λ and µ ) in (L; +)((T; +); (R; +)). Next, we will study + + (s + st + V (s + st) \ (V (s) + s 6= ;; the product of (i; x; λ) 2 Lα × Tα × Rα and (j; y; µ) 2 + + Lβ × Tβ × Rβ. Let in other words for all s; t 2 S, (s + st) D s. So D is a distributive lattice congruence on semiring S. Because (i; x; λ)(j; y; µ) = (k; z; c): + E+(S) ⊆ E•(S), each -class is a rectangular semiring. we will get that k(z; c) only depends i and j (x and y, D 0 This indicates that S is a distributive lattice of rectangular λ and µ). Let (i; x; λ) 2 Lα × Tα × Rα and (i; x ; λ) 2 S E+(S) + 0 + semiring α. Since is a normal band, the semiring Lα × Tα × Rα, then (i; x; λ) H(i; x ; λ). Actually, H is a S is a normal Clifford semiring. + 0 congruence on (S; •), so (i; x; λ)(j; y; µ) H(i; x ; λ)(j; y; µ). corollary II.1. A semiring S is a normal Clifford semiring if Let + and only if D is a distributive lattice congruence on S, every (i; x; λ)(j; y; µ) = (k; z; c); + 0 0 0 0 D-class is a rectangular semiring and E+(S) is a normal (i; x ; λ)(j; y; µ) = (k ; z ; c ): band. + 0 0 0 0 0 we have (k; z; c) (k ; z ; c ). It is clear that k = k ; c = c , With the help of the research method of theoremII.1, we H so we can see k and c are not related to x. Similarly, we can can get the following proposition: prove k and c is not related to y. Also, in fact, E+(S) is an 0 Proposition II.1. A semiring S is a left normal Clifford ideal of (S; •). So, for (i; x; λ); (i ; x; λ) 2 Lα × Tα × Rα, semiring if and only if the additive reduct (S; +) of S is (j; y; µ) 2 Lβ × Tβ × Rβ, if a left normal orthogroup in which each maximal subgroup (i; x; λ)(j; y; µ) = (k; z; c); is abelian, E+(S) ⊆ E·(S) and S satisfies the following 0 0 0 0 conditions. (i ; x; λ)(j; y; µ) = (k ; z ; c ) (1) 8s 2 S; V +(s) + s ⊇ s(s + V +(s)); through the distributive laws of S, we have (2) 8s; t 2 S; V +(st) + st ⊇ (t + V +(t))s; (3) 8s; t 2 S; V +(s) + s ⊇ s + st + V +(st)) + V +(s). (k; z; c) = (i; x; λ)(j; y; µ) 0 = ((i; 0; λ) + (i ; x; λ))(j; y; µ) 0 Next, for some distributive lattice skeleton D, let = (i; 0; λ)(j; y; µ) + (i ; x; λ)(j; y; µ) [D; L ;' ] be the strong distributive lattice D decompo- 0 0 0 α α,β = (k; 0; c) + (k ; z ; c ) sition of left normal band semiring L into left zero band 0 0 S = (k; z ; c ) semirings Lα, α2D Tα be the distributive lattice D decom- position of Clifforg semiring T into rings T , [D; R ; ] 0 0 α α α,β and hence z = z ,c = c , that is z and c are not related to i. be the strong distributive lattice D decomposition of right In the same way, through the distributive laws of S, we will normal band semiring R into right zero band semirings R , 0 α get that z and c are not related to j. For (i; x; λ); (i; x; λ ) 2 we have: Lα × Tα × Rα, (j; y; µ) 2 Lβ × Tβ × Rβ, if Theorem II.2. The spined product L× T × R = S D D (i; x; λ)(j; y; µ) = (k; z; c); α2D(Lα × Tα × Rα) of left normal band semiring L = 0 0 0 0 S (i; x; λ )(j; y; µ) = (k ; z ; c ) [D; Lα;'α,β], Clifford semiring T = α2D Tα and right normal band semiring R = [D; Rα; α,β] with respect to the in view of the distributive laws of S, we have same distributive skeleton D is a normal Clifford semiring. On the other hand, every normal Clifford semiring can be (k; z; c) = (i; x; λ)(j; y; µ) 0 expressed by such a spined product.