THE VARIETY GENERATED BY AN AI-SEMIRING OF ORDER THREE Текст научной статьи по специальности «Философия, этика, религиоведение»

URAL MATHEMATICAL JOURNAL, Vol. 6, No. 2, 2020, pp. 117-132

DOI: 10.15826/umj.2020.2.012

THE VARIETY GENERATED BY AN AI-SEMIRING

OF ORDER THREE

Xianzhong Zhao1, Miaomiao Ren1^, Sinisa Crvenkovic2, Yong Shao1, Petar DapiC2

1Northwest University, Xi'an, 710069, P.R. China 2University of Novi Sad, Novi Sad, 21102, Serbia tmiaomiaoren@yeah.net

Abstract: Up to isomorphism, there are 61 ai-semirings of order three. The finite basis problem for these semirings is investigated. This problem for 45 semirings of them is answered by some results in the literature. The remaining semirings are studied using equational logic. It is shown that with the possible exception of the semiring S7, all ai-semirings of order three are finitely based.

Keywords: Ai-semiring, Identity, Finitely based variety.

Introduction and preliminaries

By a variety we mean a class of algebras of the same type that is closed under subalgebras, homomorphic images and direct products. It is well-known (Birkhoff's theorem) that a class of algebras of the same type is a variety if and only if it is an equational class. One of the fundamental problems about a variety is the so called finite basis problem, that is, whether it can be defined by finitely many identities. If the answer is positive, then it is called finitely based. Otherwise, it is called nonfinitely based. An algebra A is said to be finitely based (resp., nonfinitely based) if the variety generated by A is finitely based (resp., nonfinitely based).

In 1951 Lyndon [9] showed that all two-element algebras are finitely based and formulated the problem whether every finite algebra is finitely based. This problem has been answered negatively, since a certain seven-element groupoid [10] was shown to be nonfinitely based. Some classical algebras are finite based. For example, so are every finite group [15], every finite associative ring [6, 8], every finite lattice [11] and every commutative semigroup [18]. However, not every finite semigroup and not every finite semiring are finitely based. The first example of an nonfinitely based finite semigroup (resp., semiring) has been given by Perkins [18] (resp., Dolinka [1]).

To seek a ultimate solution to the finite basis problem for finite algebras, Tarski [24] proposed the following problem: Is there an algorithm to decide whether a finite algebra is finitely based? McKenzie [12] negatively answered this problem for finite groupoids. However, this problem is still open when restricted to finite semigroups and finite semirings.

By a semiring we mean an algebra (S, +, ■) such that

• the additive reduct (S, +) is a commutative semigroup;

• the multiplicative reduct (S, ■) is a semigroup;

• (S, +, ■) satisfies the identities x(y + z) ~ xy + xz and (y + z)x ~ yx + zx.

One can easily find many examples of semirings in almost all branches of mathematics. Semirings can be regarded as a common generalization of both rings and distributive lattices. They have

been widely applicated in theoretical computer science and information science. We shall say that a semiring is an additively idempotent semiring (ai-semiring for short) if its additive reduct is a semilattice, i.e., a commutative idempotent semigroup. The variety of all ai-semirings is denoted by AI. Let Pf (X+) denote the set of all finite non-empty subsets of the free semigroup X + on a countably infinite set X of variables. If we define an addition and a multiplication on Pf (X+) by

A + B = A U B, A o B = {ab | a € A,b € B},

then (Pf(X+), +,o) is free in AI with respect to the mapping p: X ^ Pf(X+),x ^ {x} (see [7, Theorem 2.5]). An ai-semiring identity (Al-identity for short) over X is an expression of the

form u & v, where u,v € Pf (X+). For convenience, we write u1 + u2 +-----+ uk & v1 + v2 +-----+ vi

for the ai-semiring identity {u | 1 < i < k} & {vj | 1 < j < 1}.

In the last decades, several authors studied the finite basis problem for various semiring varieties. There is a rich literature on this subject (see [1-5, 16, 17, 19-23, 25, 27, 28]). Dolinka [1] found the first example of a finite nonfinitely based ai-semiring. In [2] he provided a sufficient condition under which an ai-semiring is inherently nonfinitely based, i.e., the variety V generated by this semiring is locally finite and every locally finite variety W for which V C W is nonfinitely based. As an application, it was shown in [3, 4] that some ai-semirings are inherently nonfinitely based1. McKenzie and Romanowska [13] showed that all ai-semirings satisfying x2 & x and xy & yx are finitely based. Zhao et al. [27, 28] considered the finite basis problem for ai-semirings satisfying x2 & x that are related to Green's relations. Based on the work of [13, 27, 28], Ghosh et al. [5] and Pastijn [16] proved that all ai-semirings satisfying x2 & x are finitely based. Ren et al. [21] showed that this result holds for all ai-semirings satisfying x3 & x. However, not every ai-semirings satisfying xn & x (n > 4) is finitely based (see [22]). Recently, Ren et al. [20] answered the finite basis problem for ai-semirings satisfying xn & x and xy & yx in which n — 1 is square-free. From these references one can find that semirings of small order have played an important role. This motivates some authors to investigate the finite basis problem for ai-semirings of small order. In this direction, Shao and Ren [23] considered the variety generated by all ai-semirings of order two. Vechtomov and Petrov [25] studied the variety generated by all semirings of order two whose multiplicative reduct is a semilattice. Moreover, McNulty and Willard [14] initiated the study of the finite basis problem for algebras of order three. The present paper follows this line of investigation. We shall systematically study the finite basis problem for ai-semirings of order three. For this, the following information about ai-semirings of order two in [23] are necessary.

Up to isomorphism, there are exactly 6 ai-semirings of order two, which are listed as L2, R2, M2, D2, N2 and T2 in Table 1. We assume that the underlying set of each of these semirings is {0,1}. Their Cayley tables for addition and multiplication are listed in the 2nd and respectively the 3rd columns of Table 1 while the 4th column contains their equational bases.

To present the solution of the word problem for ai-semirings of order two, we need to introduce the following notations. Let u be an element of X + and x an element of u. Then

o c(u) denotes the content of u, i.e., the set of all variables occurring in u.

o h(u) denotes the head of u, i.e., the first variable occurring in u.

o t(u) denotes the tail of u, i.e., the last variable occurring in u.

o l(u) denotes the length of u, i.e., is the number of variables occurring in u, where each letter is counted as many times as it occurs in u.

o m(x, u) denotes the multiplicity of x in u, i.e., the number of occurrences of x in w.

xThe semiring varieties in Dolinka's papers are types of (2, 2,0).

Table 1. The 2-element ai-semirings

Semiring Addition Multiplication Equational basis

L2 0 1 1 1 0 0 1 1 xy & x

R2 0 1 1 1 0 1 0 1 xy & y

M2 0 1 1 1 0 1 1 1 x + y & xy

D2 0 1 1 1 0 0 0 1 x2 & x, xy & yx, x + xy & x

N2 0 1 1 1 0 0 0 0 xy & zt, x + x2 & x

T2 0 1 1 1 1 1 1 1 xy & zt, x + x2 & x2

The following result follows from [23, Lemma 1.1]. We shall directly apply it without further notice.

Lemma 1. Let u & v be an nontrivial Al-identity, where u = u1 + ■ ■ ■ + uk,v = v1 + ■ ■ ■ + ve, ui,Vj € X+, 1 < i < k, 1 < j < l. Then

(i) L2 satisfies u & v if and only if {h(ui) | 1 < i < k} = {h(vj) | 1 < j < l};

(ii) R2 satisfies u & v if and only if {t(ui) | 1 < i < k} = {t(vj) | 1 < j < l};

(iii) M2 satisfies u & v if and only if IJ{c(ui) | 1 < i < k} = (J{c(vj) | 1 < j < l};

(iv) D2 satisfies u & v if and only if (Vui € u)(3vj € v)c(vj) C c(ui) and (Vvk € v)(3u^ € u)c(ui) C c(vk);

(v) N2 satisfies u & v if and only if {ui € u | l(ui) = 1} = {vj € v | l(vj) = 1};

(vi) T2 satisfies u & v if and only if {ui € u | l(ui) > 2} = 0, {vj € v | l(vj) > 2} = 0.

Up to isomorphism, there are 61 ai-semirings of order three2, which are listed as Si, 1 < i < 61 in Table 2. We assume that the carrier set of each of these semirings is {1,2,3}. Their Cayley tables for addition and multiplication are listed in Table 2. It is easy to check that there are 24 ai-semirings of order three satisfying x3 & x. By the main results of [21] we have that these semirings are all finitely based. So we only need to study the finite basis problem for the remaining 37 semirings. In fact, some of these semirings are members of the variety which are generated by all ai-semirings of order two. By the the main result of [23] it follows that they are all finitely based. Thus we have

Proposition 1. The following ai-semirings are finitely based: Si, S3, S5, S8, S9, S\0, Sii,

S12, Si3, Si4, Si5, Si6, Si7, Si8, S19, S20, S21, S22, S23, S24, S25, S26, S27, S28, S29, S30, S31, S32, S33, S34, S35, S36, S37, S38, S39, S40, S41, S42, S43, S48, S49, S50, S51, S52 and S61.

For an ai-semiring S, S* denotes the (multiplicative) left-right dual of S. It is easy to see that if S is finitely based, so is S*. Thus, in the remaining we only need to study the finite basis problem for S2, S4, S7, S44, S46, S47, S53, S55, S57, S58, S59 and S60. The following theorem is our main result.

Theorem 1. With the possible exception of S7, all ai-semirings of order three are finitely based.

2We wrote a program and obtained this result.

Table 2. The З-element ai-semirings

Semiring + Semiring +

l 1 l l 1 l l 1 l l 1 l

Sl l 2 l l l l S2 l 2 l l l l

1 1 3 1 1 1 1 1 3 1 1 2

1 1 1 1 1 1 1 1 1 1 1 1

S3 l 2 l l l l S4 l 2 l l l l

1 1 3 1 1 3 1 1 3 1 2 3

1 1 1 1 1 1 1 1 1 1 1 1

S5 l 2 l l l l S6 l 2 l l l 2

1 1 3 3 3 3 1 1 3 1 1 3

1 1 1 1 1 1 1 1 1 1 1 1

S7 l 2 l l l 2 S8 l 2 l l 2 l

1 1 3 1 2 3 1 1 3 1 1 3

1 1 1 1 1 1 1 1 1 1 1 1

S9 l 2 l l 2 l Sl0 l 2 l l 2 З

1 1 3 3 3 3 1 1 3 1 3 2

1 1 1 1 1 1 1 1 1 1 1 3

Sll l 2 l 2 2 2 Sl2 l 2 l l l З

1 1 3 3 3 3 1 1 3 1 1 3

1 1 1 1 1 3 1 1 1 1 1 3

Sl3 l 2 l l l З Sl4 l 2 l l 2 З

1 1 3 3 3 3 1 1 3 1 1 3

1 1 1 1 1 1 1 1 1 1 2 3

Sl5 l 2 l l 2 l Sl6 l 2 l l 2 З

1 1 3 3 3 3 1 1 3 1 2 3

1 1 1 2 2 2 1 1 3 1 1 1

Sl7 l 2 l 2 2 2 Sl8 l 2 З l l l

1 1 3 2 2 2 3 3 3 1 1 1

1 1 3 1 1 1 1 1 3 1 1 1

Sl9 l 2 З l l l S20 l 2 З l l l

3 3 3 1 1 3 3 3 3 3 3 3

1 1 3 1 1 1 1 1 3 1 1 1

S2l l 2 З l 2 l S22 l 2 З l 2 l

3 3 3 1 1 1 3 3 3 1 1 3

1 1 3 1 1 1 1 1 3 1 1 1

S23 l 2 З l 2 l S24 l 2 З 2 2 2

3 3 3 3 3 3 3 3 3 1 1 1

1 1 3 1 1 1 1 1 3 1 1 1

S25 l 2 З 2 2 2 S26 l 2 З 2 2 2

3 3 3 1 1 3 3 3 3 3 3 3

1 1 3 1 1 3 1 1 3 1 1 3

S27 l 2 З l l З S28 l 2 З l l З

3 3 3 1 1 3 3 3 3 3 3 3

1 1 3 1 1 3 1 1 3 1 1 3

S29 l 2 З l 2 З S30 l 2 З l 2 З

3 3 3 1 1 3 3 3 3 3 3 3

1 1 3 1 1 3 1 1 3 1 2 1

S31 1 2 3 2 2 3 S32 1 2 3 1 2 1

3 3 3 3 3 3 3 3 3 1 2 1

1 1 3 1 2 1 1 1 3 1 2 1

S33 1 2 3 1 2 1 S34 1 2 3 2 2 2

3 3 3 1 2 3 3 3 3 1 2 1

1 1 3 1 2 1 1 1 3 1 2 1

S35 1 2 3 2 2 2 S36 1 2 3 2 2 2

3 3 3 1 2 3 3 3 3 3 2 3

1 1 3 1 2 3 1 1 3 1 2 3

S37 1 2 3 1 2 3 S38 1 2 3 1 2 3

3 3 3 1 2 3 3 3 3 3 3 3

1 1 3 1 2 3 1 1 3 1 2 3

S39 1 2 3 2 2 2 S40 1 2 3 2 2 2

3 3 3 1 2 3 3 3 3 3 2 3

1 1 3 1 2 3 1 1 3 1 2 3

S41 1 2 3 2 2 2 S42 1 2 3 2 2 3

3 3 3 3 3 3 3 3 3 3 2 3

1 1 3 1 2 3 1 1 3 2 2 1

S43 1 2 3 2 2 3 S44 1 2 3 2 2 2

3 3 3 3 3 3 3 3 3 1 2 3

1 1 3 2 2 1 1 1 3 2 2 2

S45 1 2 3 2 2 2 S46 1 2 3 2 2 2

3 3 3 2 2 3 3 3 3 1 2 3

1 1 3 2 2 2 1 1 3 2 2 2

S47 1 2 3 2 2 2 S48 1 2 3 2 2 2

3 3 3 2 2 1 3 3 3 2 2 2

1 1 3 2 2 2 1 1 3 2 2 2

S49 1 2 3 2 2 2 S50 1 2 3 2 2 2

3 3 3 2 2 3 3 3 3 3 3 3

1 1 3 2 2 3 1 1 3 2 2 3

S51 1 2 3 2 2 3 S52 1 2 3 2 2 3

3 3 3 2 2 3 3 3 3 3 3 3

1 1 3 3 1 3 1 1 3 3 1 3

S53 1 2 3 1 2 3 S54 1 2 3 3 2 3

3 3 3 3 3 3 3 3 3 3 3 3

1 1 3 3 2 3 1 1 3 3 2 3

S55 1 2 3 2 2 2 S56 1 2 3 3 2 3

3 3 3 3 2 3 3 3 3 3 2 3

1 1 3 3 3 3 1 1 3 3 3 3

S57 1 2 3 1 2 3 S58 1 2 3 2 2 2

3 3 3 3 3 3 3 3 3 3 3 3

1 1 3 3 3 3 1 1 3 3 3 3

S59 1 2 3 3 1 3 S60 1 2 3 3 2 3

3 3 3 3 3 3 3 3 3 3 3 3

1 1 3 3 3 3

S61 1 2 3 3 3 3

3 3 3 3 3 3

1. The proof of Theorem 1

In this section we shall provide the proof of Theorem 1. Let HSP(S) denote the variety generated by an ai-semiring S and к the set {1,2,..., k} for a positive integer k. We start with a technique that will be used repeatedly in the sequel. Suppose that £ is a set of identities which include the identities that determine AI and that u w v is an Al-identity, where u =

«! + •••+ Uk,v = v\ + • • • + V£,Ui,Vj € X+,i € k,j € £. Then it is easy to see that the ai-semring variety defined by u w v is equal to the ai-semiring variety defined by the simpler identities и pa и + Vj,v & v + in, i € k,j € £. Thus, to show that и ~ v is derivable from E, we only need to show that и ~ и + vj,v & v + щ, i € k, j € £ can be derived from E.

Proposition 2. HSP(S2) is the ai-semiring variety determined by the identities

XiX2X3 w У1У2У3, (1.1)

x + x2 w x3, (1.2)

x2 + y2 w xy, (1.3)

x3 + y w x3. (1.4)

Proof. An Al-term is said to be in canonical form if it is equal to one of the following

terms: xi + ■ ■ ■ + xm, x2 + ■ ■ ■ + x^, xi +----+ xm + y2 + ■ ■ ■ + y^ and x3, where xi,..., x^^ are

distinct, variables, yi,...,yn are distinct variables, and {xi \ i € m}f){yj \ j € n} = 0. Suppose that и = щ + U2 + • • • + Uk is an Al-term, where гн € X+, i e k. We shall show that there exists an Al-term u' in canonical form such that the identities (1.1)-(1.4) and the identities determining AI imply the identity u w u'. The following cases are needed.

£(ih) = 1 for all i € k. Then u = x\ + • • • + x. £(ih) = 2 for all i € k. Then the identity (1.3) implies u & x\ + • • • + x2

m

£(u,i) < 2 for all i € k, £(it.¿J = 1 for some i\ € k and £(ih2) = 2 for some ¿2 € k. If c(ui^ c(uj) = 0 for some ui and uj with l(ui) = 1 and l(uj) = 2, then the identities (1.2)-(1.4) implies u & x3. Otherwise, we have that the identity (1.3) implies u & x1 + ■ ■ ■ + xm +

2/i H-----1- Vn, where {xi \ i € m} f|{yj \ j € n} = 0.

• £(ui) > 3 for some i € k. Then the identities (1.1) and (1.4) imply u & x3.

It is routine to check that S2 satisfies the identities (1.1)-(1.4). In the remainder we shall show that every identity which is satisfied in S2 can be derived from the identities (1.1)-(1.4) and the identities determining Al. By the above arguments it is enough to show that if S2 satisfies an identity u & v, where u and v are Al-terms in canonical forms, then the identities (1.1)-(1.4) and the identities determining Al imply u & v. Notice that T2 can be embedded into S2. We only need to consider the following cases:

• u = x1 + ■ ■ ■ + xm, v = y1 + ■ ■ ■ + yn. It is easy to see that u & v is trivial.

• u = x\ + ■ ■ ■ + x^n, v = y2 + ■ ■ ■ + yn. It is easy to see that u & v is trivial.

• u = x2 + ■ ■ ■ + x'm, v = y1 + ■ ■ ■ + yk + z2 + ■ ■ ■ + z|. Let y : Pf (X+) ^ S2 be a semiring homomorphism such that <p(x) = 3 for every variable x in X. Then <p(u) = 2 and y(v) = 1, a contradiction. Thus u & v is not satisfied in S2.

• u = xf + ■ ■ ■ + x'm,v = t3. This case is similar to the preceding one.

• u = yi +----+ yk + zf + ■ ■ ■ + z2 ,v = y' +----+ y'm + z'2 + ■ ■ ■ + z'2n. It is easy to see that

U tt v is trivial.

• u = y1 +-----+ yk + zf +----+ zf, v = x3. Let ^ : Pf (X+) — S2 be a semiring homomorphism

such that tp(yi) = 2, <p(zj) = 3 and <p(t) = 1 for all i € k, j € £ Then <£>(«) = 2 and <p(v) = 1, a contradiction. Thus u tt v is not satisfied in S2.

• u = xi,v = x3. Then the identity (1) implies u tt v.

This completes the proof. □

Proposition 3. HSP(S4) is the ai-semiring variety determined by the identities

xy tt x2y, (1.5)

xyz tt yxz, (1.6)

x + y2 tt xy2, (1.7)

x + yz tt yx + yz. (1.8)

Proof. An Al-term is said to be in canonical form if it is equal to one of the following terms:

x1 +-----+ xm, xi ■ ■ ■ xm and xi ■ ■ ■ xm(yi +-----+ yn), where x1,..., xm are distinct variables, y1,...,yn

are distinct variables and {xi | i € m} f]{yj | j € n} = 0. Suppose that u = u\ +«2 H-----1is an

Al-term, where Ui € X+, i e k. We shall show that there exists an Al-term u' in canonical form such that the identities (1.5)-(1.8) and the identities determining AI imply the identity u tt u'. The following cases are needed.

• £(ih) = 1 for all i € k. Then u = x\ + • • • + xm.

• m(t(ui),ui) > 2 for some j € k. Then the identities (1.5)—(1.7) imply u tt x2 ■ ■ ■ xm.

• £(ui) > 2 for some i € k, m(t(uj),uj) = 1 for every j e k. Then the identities (1.5), (1.6) and (1.8) imply u ~ x2 • • • x'm(yi + • • • + yn), where {xi \ i € m} CliUj I j ^ 11} = 0.

It is routine to check that S4 satisfies the identities (1.5)-(1.8). By the above arguments it is enough to show that if S4 satisfies an identity u tt v, where u and v are Al-terms in canonical forms, then the identities (1.5)-(1.8) and the identities determining AI imply u tt v. Since T2 can be embedded into S4, we only need to consider the following cases:

• u = x1 + ■ ■ ■ + xm, v = y1 + ■ ■ ■ + yn. It is easy to see that u tt v is trivial.

• u = x2 ■ ■ ■ xm, v = y2 ■ ■ ■ yn. It is easy to see that u tt v is trivial.

• u = x2 ■ ■ ■ xm, v = y2 ■ ■ ■ y2(z1 +-----+ zi). Let <p : Pf (X+) — S4 be a semiring homomorphism

such that tp(yi) = 3, <p(zj) = 2, i € k, j € £, <p(x) = 1 for every remaining variable x. Then p(u) = 1 or 3, p(v) = 2, a contradiction. Thus u tt v is not satisfied in S4.

• u = x2 ■ ■ ■ xm(y1 +----+ yn), v = z2 ■ ■ ■ zf(t1 + ■ ■ ■ + ti). It is easy to see that u tt v is trivial.

This completes the proof. □

Proposition 4. HSP(S44) is the ai-semiring variety determined by the identities

x3 tt x2, (1.9)

xy tt yx, (1.10)

x + xy tt x, (1.11)

x2y + xy2 tt xy. (1.12)

Proof. An Al-term u = u1 + ■ ■ ■ + un is said to be in canonical form if every ui is equal to one of the following terms: x, x- ■ ■ ■ x2m and x- ■ ■ ■ x2my, where xi,..., x^^ are distinct variables and y / Xi for every i € m. Let p = xi • • • xn be an element of X+ such that n > 2. By induction on

n we have that the identity (1.10) and (1.12) imply p && ^ ^l^i^n xi ' ' ' xi_xxi+1 ' ' ' xnxi. It follows

that for any Al-term u, there exists an Al-term u' in canonical form such that (1.9)-(1.12) imply u & u'.

It is easy to check that S44 satisfies the identities (1.9)-(1.12). To show that HSP(S44) is determined by (1.9)-(1.12), by the above arguments it suffices to show that if S44 satisfies u+p & u, where u+p and u are Al-terms in canonical forms, then the identities (1.9)-(1.12) and the identities determining AI imply u + p & u. We shall consider the following three cases.

• p = x. Since N2 can be embedded into S44, there exists some ui in u such that ui = x. It follows that u + p & u is trivial.

• p = x- ••• xm. Since D2 can be embedded into S44, there exists some ui in u such that c(ui) C c(p) and so (1.9) and (1.10) imply uip & p. Further, we have

(1.11)

u & u + ui & u + ui + uip & u + ui + p & u + p.

• p = xl ■ ■ ■ xmy. Since D2 can be embedded into S44, we have that {ui € u | c(ui) C c(p)} is non-empty. Suppose that for any ui in {ui € u | c(ui) C c(p)}, there exists x in c(ui) such that m(x,p) < m(x,ui). That is to say, m(y,ui) = 2 for every ui in {ui | c(ui) C c(p)}. Let f : Pf (X+) ^ S44 be a semiring homomorphism such that (p(z) =2 for every z € X \ c(p), cp(xi) = 3 for every i e m and <p(y) = 1. Then tp(u) = 2 and <p(u + p) = 1, a contradiction. Thus there exists ui in {ui € u | c(ui) C c(p)} such that m(x,ui) < m(x,p) for every x in c(ui). If y € c(ui), then m(y, ui) = 1 and so (1.9) and (1.10) imply uix1 ■ ■ ■ xjn & p. Further, we have

(1.11) 2 2 u & u + ui & u + ui + uix-y ••• xm & u + p.

If y / c(ui), then (1.9) and (1.10) imply uip & p. We therefore have

(1.11)

u & u + ui & u + ui + upp & u + p. This completes the proof. □

Proposition 5. HSP(S46) is the ai-semiring variety determined by the identities

x2y & xy, (1.13)

2 2 2 2 x y & y x , (1.14)

xyz & yxz, (1.15)

x + xy & x, (1.16)

x + yx & x. (1.17)

Proof. An Al-term u = u1 + ■ ■ ■ + un is said to be in canonical form if every ui is equal to one of the following terms: x, x\ ■ ■ ■ and x\ ■ ■ ■ x2%y, where y / Xi for all i € m. Let p be an element of X+ such that l(p) > 2. If m(t(p),p) = 1, then the identities (1.13)-(1.15) imply p & xl ■ ■ ■ xmy. If m(t(p),p) > 2, then the identities (1.13)-(1.15) imply p & xl ■ ■ ■ xm. It follows that for any Al-term u, there exists an Al-term u' in canonical form such that (1.13)-(1.15) imply u & u'.

It is routine to check that S46 satisfies the identities (1.13)-(1.17). To show that HSP(S46) is the ai-semiring variety determined by (1.13)-(1.17), by the above arguments it suffices to show that if S46 satisfies u+p œ u, where u + p and u are AI-terms in canonical form, then (1.13)-(1.17) imply u + p œ u. The following three cases are necessary.

• p = x. Since D2 can be embedded into S46, there exists some u in u such that c(u) = {x}. Suppose that u = x2 for every u in u with c(u) = {x}. Let <p : X ^ S46 be a semiring homomorphism such that f (x) = 1 and f(y) = 2 for every y = x. Then f (u) = 2 and <p(u + p) = 1, a contradiction. Thus there exists uj in u such that u = x and so u + p œ u is trivial.

• p = x2 ••• xm. Since D2 can be embedded into S46, there exists some uj in u such that c(uj) Ç c(p) and so (1.13)-(1.15) imply p œ ujp. We now have

(1.16)

u + p œ u + uj + p œ u + uj + up œ u + uj œ u.

• p = x1 ■ ■ ■ x^my. Since D2 can be embedded into S46, it follows that {uj € u | c(uj) Ç c(p)} is non-empty. Suppose that m(y,uj,) = 2 for every uj in {uj € u | c(uj) Ç c(p)}. Let f : Pf (X+) ^ S46 be a semiring homomorphism such that <p(z) = 2 for every z / c(p), (p(xi) = 3 for every i e m and <p(y) = 1. Then = 2 and + p) = 1, a contradiction. Thus we only need to consider the following cases:

o (3uj € {uj € u | c(uj) Ç c(p)}) y / c(uj). Then

(1.13)-(1.15) (1.16)

u + p œ u + uj + p œ u + uj + u^ œ u + uj œ u.

o (3uj € {uj € u | c(uj) Ç c(p)}) y € c(uj), t(uj) = y and m(y,uj) = 1. Then

(1.13)-(1.15) 2 2 (1.17)

u + p œ u + uj + p œ u + uj + x1 ••• x,^^ œ u + uj œ u.

This completes the proof. □

Proposition 6. HSP(S47) is the ai-semiring variety determined by the identities

xy œ yx, (1.18)

x + xy œ x, (1.19)

x2 + xy œ x2, (1.20)

x + x1x2x3 œ x. (1.21)

Proof. It is easy to verify that S47 satisfies the identities (1.18)-(1.21). In the remainder it suffices to show that every identity which is satisfied in S47 is derivable from (1.18)-(1.21). Let

u+p & u be such an identity, where u = u\ H-----H/,m, iH,p € X+,i € m. We consider the following

three cases.

• l(p) = 1. Since N2 can be embedded into S47, there exists ui in u such that ui = p. Thus u + p & u is trivial.

• l(p) = 2. Suppose that for any щ in u, е(щ) ^ c(p). Let f: Pf (X+) — S47 be a semiring homomorphism such that f(z) = 2 for every z € X \ c(p) and f(x) = 3 for every x € c(p). Then f(u) = 2 and f(u + p) = 1, a contradiction. Thus there exists щ in u such that c(ui) С c(p). Assume that 1(щ) > 3 for every u in (u € u | c(u) С c(p)}. Then f(u) = 2 and f (u+p) = 1, a contradiction. This implies that there exists uj in u such that c(uj) С c(p) and l(uj) < 2. Further, the identities (1.18)-(1.20) imply

u + p & u + uj + p & u + uj & u.

• l(p) > 3. Then u + p & u can be derived from (1.21).

This completes the proof. □

Proposition 7. HSP(S53) is the ai-semiring variety determined by the identities

xy & yx, (1.22)

xy + y2 & x + y2, (1.23)

x + xy & xy, (1.24)

xy + yz + xz & xyz. (1.25)

Proof. An Al-term is said to be in canonical form if it is equal to one of the following terms:

ЖН-----\-xm, Xiyi H-----bXmym, and Xi H-----Krm + н-----\-ynZn, where {xi \ i € m} f|{yj, zj \

j € n} = 0, x'2 and xy can not occur simultaneously. Suppose that и = щ + U2 + • • • + и к is an Al-term, where щ € X+,i e k. It is easy to show that there exists an Al-term u' in canonical form such that the identities (1.22)-(1.25) imply the identity u & u .

It is routine to check that S53 satisfies the identities (1.22)-(1.25). In the remainder we shall show that every identity which is satisfied in S53 can be derived from the identities (1.22)-(1.25). By the above arguments it suffices to show that if S53 satisfies an identity u & v, where u and v are Al-terms in canonical form, then the identities (1.22)-(1.25) and the identities determining AI imply u & v. Notice that T2 can be embedded into S53. We only need to consider the following cases:

• u = xi + ■ ■ ■ + xm, v = yi + ■ ■ ■ + yn. It follows immediately that u & v is trivial.

• и = х\у\ + • • • + xmym, v = z\S\ + • • • + znsn. For any i € m, suppose that {.Ti,y\} is not equal to {zj, Sj} for every j € n. Consider the following two subcases:

^ xj = yj. Let (f : Pf (X+) — S53 be a semiring homomorphism such that ^(xj) = 1 and ^>(z) = 2 for every z € X \ (xj}. Then f(u) = 3 and f(v) = 1, a contradiction.

^ xj = yj. Since M2 can be embedded into S53, we can deduce that S53 satisfies one of the following identities: xjyj & x2 + y2, xjyj & x2 + yj, xjyj & xj + y2 and xjyj & xj + yj, a contradiction.

Thus {xi, yi} is equal to {zj, Sj} for some j € n. Similarly, for any j € n, {zj, Sj} is equal to {xi, yi} for some i € m. Hence и рз v is trivial.

• u = x1y1 + ■ ■ ■ + xmym, v = z1 + ■ ■ ■ + zk + s1t1 + ■ ■ ■ + site. We consider the following two subcases.

о {xi^yi} ^ izi I G k} for some i € m. Let tp : Pf(X+) —>• S53 be a semiring homomorphism such that ip(zi) = 1 for every i € к and tp(x) = 2 for every z € X \ {zi | i € к}. Then ф(ч) = 3 and ^(v) = 1, a contradiction.

o {xiiUi} ^ izi I G t} for every i e m. Notice that M2 can be embedded into S53. Let d : Pf (X+) ^ S53 be a semiring homomorphism such that d(xi) = d(yi) = 1 if {xiiUi} f]{zi I ^ k} / 0, and 9(y) = 2 for every remaining variable y. Then 9(u) = 3 and d(v) = 1, a contradiction.

This shows that u& v is not satisfied in S53.

• u = xi +----+ xm + yizi +----+ ynZn, V = xl + ■ ■ ■ + x'k + ylzl + ■ ■ ■ + y'(z'f . Suppose that

x1 = yi for some i.

o yi = zi. Choose every variable in X \ {x1} to 2. Then S53 satisfies x1 & xl, a contradiction.

o yi = zi. Choose every variable in X \ {yi, z-} to 2. Then S53 satisfies yi + zi & yizi or yi + zi1 & yizi, a contradiction.

This implies that x1 +-----+ xm & x1 +-----+ x'k is trivial and so S53 satisfies y1z1 +-----+ ynzn &

y1z1 + ■ ■ ■ + y'ez'e. By the preceding case it follows that y1z1 + ■ ■ ■ + ynzn & y1z1 + ■ ■ ■ + y'fz'e is trivial. Hence u& v is trivial.

This completes the proof. □

Proposition 8. HSP(S55) is the ai-semiring variety determined by the identities

Proof. It is routine to check that S55 satisfies (1.26)-(1.29). In the remainder it suffices to

show that if S55 satisfies u ~ u+q, where u = Ui+U2~\-----\-um, ih, q € X+, i e m, then (1.26)-(1.29)

and the identities determining AI imply the identity u & u + q. Choose Z = (Ui£m c(ui))\c(q). By [21, Lemma 2.11] we have that T2 satisfies Dz{u) ~ Dz{u) + q, where Dz{u) denotes the sum of terms ui for which c(ui) C c(q). We may assume that DZ(u) = u1 + u2 + ■ ■ ■ + uk. The following two cases are necessary.

• l(q) = 1. Then (1.27) and (1.29) implies u & u + q.

• l(q) > 2. Then there exists ui with c(ui) C c(q) such that l(ui) > 2. Further, by (1.26)-(1.28)

xy & yx, xy & x y, xy & xy + xyz,

2 2 1

x & x + x.

(1.26)

(1.27)

(1.28) (1.29)

we have

(1.28)

(1.26),(1.27)

u & u + ui

u + u- + u-q

u + ui + q & u + q.

This completes the proof.

□

Proposition 9. HSP(S57) is the ai-semiring variety determined by the identities

xyz & yxz, x 1 y & xy, x + yz & yx + yz,

o

x 1 + xy & xy.

(1.30)

(1.31)

(1.32)

(1.33)

Proof. An AI-term is said to be in canonical form if it is equal to x1 + ■ ■ ■ + xm, x 1 or

x\ - ■ ■ xm(yi H-----h yn), where {xi \ i € m} f]{yj \ j € n} = 0. Let u be an arbitrary AI-term. It is

easy to see that there exists an AI-term u' in canonical form such that (1.30)-(1.33) imply u & u'.

It is routine to check that S57 satisfies (1.30)-(1.33). In the remainder it is enough to show that if S57 satisfies u & v where u and v are AI-terms in canonical form, then (1.30)-(1.33) and the identities determining AI imply u & v. Notice that both M1 and T2 can be embedded into S57. We consider the following nontrivial case that u = x1 ■ ■ ■ xm(y1 + ■ ■ ■ + yn), v = z1 ■ ■ ■ zk(t1 + ■ ■ ■ + tg). For a fixed Xi, suppose that it is not equal to Zj for every j € k. Since M2 can be embedded into S57, it follows that xi must be equal to some tj. Choose xi to 1 and every other variable to 2. We have that 3 = 1, a contradiction. Thus x 1 ■ ■ ■ xm && z1 ■■■ zk is trivial. Choose x- to 2 for every i € m. Then S57 satisfies y\ + • • • + yn & t\ + • • • + tg. Thus y\ + • • • + yn & t\ + • • • + tg is trivial and so is u & v. □

Proposition 10. HSP(S58) is the ai-semiring variety determined by the identities

Proof. An AI-term is said to be in canonical form if it is equal to x1 +-----+ xm, y- +-----+ yn

or x\ + • • • + xm + y'l + • • • + y.2, where {xi \ i € m} f]{yj | i € n} = 0. Let u be an arbitrary AI-term. It is easy to see that there exists an AI-term u' in canonical form such that (1.34) and (1.35) imply u & u'.

It is routine to check that S58 satisfies (1.34) and (1.35). In the remainder it is enough to show that if S58 satisfies u & v, where u and v are AI-terms in canonical form, then (1.34), (1.35) and the identities determining AI imply u & v. Notice that T2 can be embedded into S58. The following two cases are necessary.

• u = x1 + ■ ■ ■ + xm, v = y1 + ■ ■ ■ + yn. Then u & v is trivial.

• u = x1 + ■ ■ ■ + xm, v = y1 + ■ ■ ■ + yn. Since L2 can be embedded into S58, it follows that u& v is trivial.

• u = x1 + ■ ■ ■ + xk, v = y1 + ■ ■ ■ ym + z1 + ■ ■ ■ + z2n. Let ^ : Pf (X+) ^ S58 be a semiring homomorphism such that = 1 for every i e m and ip(x) = 2 for every remaining variable x. Since L1 can be embedded into S58, it follows that ^(u) = 3 and ^(v) = 1, a contradiction.

• u = x1 + ■ ■ ■ xm + y1 + ■ ■ ■ + yn, v = z1 + ■ ■ ■ zk + t1 + ■ ■ ■ + t1. Since L1 can be embedded into S58, we have

For a fixed Xi, suppose that it is not equal to Zj for every j e k. Then Xi must be equal to some tj. Choose xi to 1 and every remaining variable to 2. We have that 1 = 3, a contradiction. Thus Xi is equal to some Zi and so {xi \ i € m} = {zj \ j € k}. Hence u ~ v is trivial.

xy & x ,

11 x & x + x .

,2

(1.34)

(1.35)

{Xi I i e m}\J{yi I i en} = {Zj I j ek} |J{ij | j e £}.

This completes the proof.

□

The Variety Generated by an Ai-Semiring of Order Three 129 Proposition 11. HSP(S59) is the ai-semiring variety determined by the identities

£1x2x3 — V1V2V3, (1-36)

x3 + v — x3, (1.37)

x2 + v2 - xy, (1.38)

x + x2 - x2. (1.39)

Proof. An Al-term is said to be in canonical form if it is equal to x3, x1 + ■ ■ ■ + xm,

yf H----+ y2 or xi H----+ xm + y'l H----+ where {xi | i € m} f|{yj | j 6 n} = 0. Let u be an

arbitrary Al-term. It is easy to see that there exists an Al-term u' in canonical form such that (1.36)-(1.39) imply u — u'.

It is routine to check that S59 satisfies (1.36)-(1.39). In the remainder it is enough to show that if S59 satisfies u — v, where u and v are Al-terms in canonical form, then (1.36)-(1.39) and the identities determining AI imply u — v. Notice that T2 can be embedded into S59. The following cases are necessary.

• u = x1 + ■ ■ ■ + xm, v = y1 + ■ ■ ■ + Vn- It is easy to see that u — v is trivial.

• u = tl, v = t2. Then (1.36) implies u — v.

• u = t3, v = v2 +-----+ yn. Choose every variable to 2. Then 3 = 1, a contradiction. Thus S59

does not satisfy u — v.

• u = t3, v = xi + ••• + xm

+ V2 +----+ Vn. Choose every variable to 2. Then 3 = 1, a

contradiction. Thus S59 does not satisfy u — v.

• u = xi + ■ ■ ■ + xm, v = y2 + ■ ■ ■ + yn. It is easy to see that u — v is trivial.

• u = xi + ■ ■ ■ + xm, v = y1 + ■ ■ ■ + yk + zf + ■ ■ ■ + z|. Consider the following two subcases.

o {yi | i € k} ^ {xi | i € m}. Choose yi to 3, where yi £ {xi \ i € m}. Choose every other variable to 2. Then 1=3, a contradiction.

o {yi | i € k} C {xi | i € m}. Choose yi to 1 for every i e m and every other variable to 2. Then 3=1, a contradiction.

Thus S59 does not satisfy u — v.

• u = x1 + ■ ■ ■ + xm + v2 + ■ ■ ■ + yn, v = z1 + ■ ■ ■ + zk + s1 + ■ ■ ■ + s2. Fix xi. Suppose that xi is not equal to Zj for every j € k.

o Xi € {sj | j € Choose x^ to 1 and every other variable to 2. Then 1=3, a contradiction.

o Xi £ {sj | j € Choose x^ to 3 and every other variable to 2. Then 3=1, a contradiction.

Thus Xi is equal to Zj for some j € k and so xi H----+ xm z\ +----h Zk is trivial. Rirt.her,

y2 + ■ ■ ■ + y2^ — s1 + ■ ■ ■ + sj holds in S59. By the case 5 we have that this identity is trivial and so is u — v.

This complete the proof.

□

Proposition 12. HSP(S60) is the ai-semiring variety determined by the identities

x3 « x2,

2 i 2 x + y « xy,

o 9

x + x 2 « x 2.

,2

,2

,2

(1.40)

(1.41)

(1.42)

Proof. An AI-term is said to be in canonical form if it is equal to x1 +-----h xm, y 1 +-----+ yn

or x\ + • • • + xm + y'l + • • • + y.2, where {xi | i € m} f]{yj | j € n} = 0. Let t/, be an arbitrary AI-term. It is easy to see that there exists an AI-term u in canonical form such that (1.40)-(1.42) imply u & u .

It is routine to check that Seo satisfies (1.40)-(1.42). In the remainder it suffices to show that if Seo satisfies u & v, where u and v are terms in canonical form, then (1.40)-(1.42) and the identities determining AI imply u & v. Notice that T1 can be embedded into Seo. The following cases are necessary.

• u = x1 + ■ ■ ■ + xm, v = y1 + ■ ■ ■ + yn. It is easy to see that u & v is trivial.

• u = x1 + ■ ■ ■ + xm, v = y 1 + ■ ■ ■ + yn. Since M1 can be embedded into S60, it follows that u& v is trivial.

• u = x\ H----+x'm,v = yH-----\-yk + zi H----+ zj. Then {yi | i e k} C {xj | i € m}. Choose

yi to 1 for every i e k and every other variable to 2. Then 3=1, a contradiction. Thus ¿>60 does not satisfy u & v.

• u = x1 +-----+ xm + y 1 +-----+ yn, v = z1 +-----+ zk + s 1 +-----+ s"1. Then

{Xi I i e rn}\J{yi I i Gn} = {Zj I j Gk} |J{Si I j € l).

Fix Xi. Suppose that Xi is not eciual to Zj for all j € k. Choose Xi to 1 for all i € k and every other variable to 2. Then 1=3, a contradiction. Thus Xi = Zj for some j € k. This implies that u& v is trivial.

This complete the proof. □

By Propositions 1-12 we immediately complete the proof of Theorem 1.

We have answered the finite basis problem for all ai-semirings of order three except S7. This will lay a solid foundation for our subsequent work about ai-semiring varieties. Moreover, we conjecture that the semiring S7 is nonfinitely based. In contrast to the rich results in the theory of semigroup varieties [26], there are still many problems to be solved in the theory of semiring varieties. In particular, it is of the interest to study the variety generated by all ai-semirings of order three.

The authors are particularly grateful to Professor Mikhail V. Volkov for his helpful suggestions contributed to this paper. The authors also thank the anonymous referees for their valuable comments and suggestions which lead to the final version of this paper. Xianzhong Zhao is supported by National Natural Science Foundation of China (11971383, 11571278). Miaomiao Ren, corresponding author, is supported by National Natural Science Foundation of China (11701449). Yong Shao is supported by Natural Science Foundation of Shaanxi Province (2020JM-425).

2. Conclusion

Acknowledgements

REFERENCES

1. Dolinka I. A nonfintely based finite semiring. Int. J. Algebra Comput., 2007. Vol. 17, No. 8. P. 1537-1551. DOI: 10.1142/S0218196707004177

2. Dolinka I. A class of inherently nonfinitely based semirings. Algebra Universalis, 2009. Vol. 60, No. 1. P. 19-35. DOI: 10.1007/s00012-008-2084-y

3. Dolinka I. The finite basis problem for endomorphism semirings of finite semilattices with zero. Algebra Universalis, 2009. Vol. 61, No. 3-4. P. 441-448. DOI: 10.1007/s00012-009-0024-0

4. Dolinka I. A remark on nonfinitely based semirings. Semigroup Forum, 2009. Vol. 78, No. 2. P. 368-373. DOI: 10.1007/s00233-008-9096-y

5. Ghosh S., Pastijn F., Zhao X. Z. Varieties generated by ordered bands I. Order, 2005. Vol. 22, No. 2. P. 109-128. DOI: 10.1007/s11083-005-9011-z

6. Kruse R.L. Identities satisfied by a finite ring. J. Algebra, 1973. Vol. 26, No. 2. P. 298-318. DOI: 10.1016/0021-8693(73)90025-2

7. Kuril M., Polak L. On varieties of semilattice-ordered semigroups. Semigroup Forum, 2005. Vol. 71, No. 1. P. 27-48. DOI: 10.1007/s00233-004-0176-3

8. L'vov I.V. Varieties of associative rings. I. Algebra and Logic, 1973. Vol. 12, No. 3. P. 150-167. DOI: 10.1007/BF02218695

9. Lyndon R. C. Identities in two-valued calculi. Trans. Amer. Math. Soc., 1951. Vol. 71, No. 3. P. 457-457. DOI: 10.1090/S0002-9947-1951-0044470-3

10. Lyndon R. C. Identities in finite algebras. Proc. Amer. Math. Soc., 1954. Vol. 5. P. 8-9. DOI: 10.1090/S0002-9939-1954-0060482-6

11. McKenzie R. Equational bases for lattice theories. Math. Scand., 1970. Vol. 27. P. 24-38. DOI: 10.7146/math.scand.a-10984

12. McKenzie R. Tarski's finite basis problem is undecidable. Int. J. Algebra Comput., 1996. Vol. 6, No. 1. P. 49-104. DOI: 10.1142/S0218196796000040

13. McKenzie R. C., Romanowska A. Varieties of -distributive bisemilattices. Contrib. Gen. Algebra, 1979. Vol. 1. P. 213-218.

14. McNulty G.F., Willard R. The Chautauqua Problem, Tarski's Finite Basis Problem, and Residual Bounds for 3-element Algebras. In progress.

15. Oates S., Powell M.B. Identical relations in finite groups. J. Algebra, 1964. Vol. 1, No. 1. P. 11-39. DOI: 10.1016/0021-8693(64)90004-3

16. Pastijn F. Varieties generated by ordered bands II. Order, 2005. Vol. 22, No. 2. P. 129-143. DOI: 10.1007/s11083-005-9013-x

17. Pastijn F., Zhao X. Z. Varieties of idempotent semirings with commutative addition. Algebra Universalis, 2005. Vol. 54, No. 3. P. 301-321. DOI: 10.1007/s00012-005-1947-8

18. Perkins P. Bases for equational theories of semigroups. J. Algebra, 1969. Vol. 11, No. 2. P. 298-314. DOI: 10.1016/0021-8693(69)90058-1

19. Ren M. M., Zhao X. Z. The varieties of semilattice-ordered semigroups satisfying x3 — x and xy — yx. Pernod. Math. Hungar., 2016. Vol. 72, No. 2. P. 158-170. DOI: 10.1007/s10998-016-0116-5

20. Ren M.M., Zhao X. Z., Shao Y. The lattice of ai-semiring varieties satisfying xn — x and xy — yx. Semigroup Forum, 2020. Vol. 100, No. 2. P. 542-567. DOI: 10.1007/s00233-020-10092-8

21. Ren M. M., Zhao X. Z., Wang A. F. On the varieties of ai-semirings satisfying x3 — x. Algebra Universalis, 2017. Vol. 77, No. 4. P. 395-408. DOI: 10.1007/s00012-017-0438-z

22. Ren M. M., Zhao X. Z., Volkov M. V. The Burnside Ai-Semiring Variety Defined by xn — x. Manuscript.

23. Shao Y., Ren M. M. On the varieties generated by ai-semirings of order two. Semigroup Forum, 2015. Vol. 91, No. 1. P. 171-184. DOI: 10.1007/s00233-014-9667-z

24. Tarski A. Equational logic and equational theories of algebras. Stud. Logic Found. Math., 1968. Vol. 50. P. 275-288. DOI: 10.1016/S0049-237X(08)70531-7

25. Vechtomov E. M., Petrov A. A. Multiplicatively idempotent semirings. J. Math. Sci., 2015. Vol. 206, No. 6. P. 634-653. DOI: 10.1007/s10958-015-2340-6

26. Volkov M.V. The finite basis problem for finite semigroups. Sci. Math. Jpn., 2001. Vol. 53, No. 1. 171-199.

27. Zhao X. Z., Guo Y. Q., Shum K.P. D-subvarieties of the variety of idempotent semirings. Algebra Colloquium, 2002. Vol. 9, No. 1. P. 15-28.

28. Zhao X. Z., Shum K.P., Guo Y. Q. L-subvarieties of the variety of idempotent semirings. Algebra Universalis, 2001. Vol. 46, No. 1-2. P. 75-96. DOI: 10.1007/PL00000348