Homological dimension of Rees matrix semigroups in intelligent fuzzy systems

1 Introduction

Let Y be a left R-module. The projective dimension pd(Y) is a minimum integer n in order that there exists a resolution of Y by projective modules. 0 → X n → ⋯ → X 1 → X 0 → Y → 0 .

The definitions of the injective dimension id (Y) and flat dimension fd(Y) are given Similarly. If no finite resolution exists, we set pd(Y), id(Y) or fd(Y) equal to.

Global Dimension Theorem the numbers are the same in the following situations for any ring R:

sup {id(Y): Y any left R-module}

This common number (possibly^∞) is called the (left) global dimension of R, which is described by l.gl.dim R. Bourbaki calls it the homological dimension.

In the similar way, we have the definition of the right global dimension. Assume that R is a commutative ring, l.gl.dim R=r.gl.dim R. Clearly l.gl.dim R=0 means that ring R is both semisimple and Artinian.

The upper bound of l.gl.dim K[S] is given, where S is a finite regular semigroup and ring K is commutative (Nico. W.R.1971, 1972). In (Kuzmanovich. J., Teply. M.L. 1997), Kuzmanovich, etc. discussed bounds of l.gl.dim K[S], where S is a monoid with a series of ideals in order that every factor is a finite nonnull Rees matrix semigroup. Jespers and Wang discovered an upper bound of the l.gl.dim of the contracted semigroup algebra, where the monoid has a chain of ideals which factors are nonnull Rees matrix semigroups (Jespers. E., Wang. Q. 2000). S.Margolis, B.Steinberg applied homological methods to exactly obtain the quiver of a right regular band (Margolis. S., Steinberg. B. 2011).

In this paper, for S a finite monoid, first we give some characterizations of Rees matrix semigroup algebras. Then if K is a ring and S has a Rees matrix ideal M, we give the bounds of r.gl.dim K[S] in terms of r.gl.dim K [T _i ] and r.gl.dim K[S/M]. If a finite monoid S is a 0-direct union of some finite Rees matrix semigroups, which are in the forms of S _i  = M (T _i  ; I _i  ; Λ _i  ; P _i ), we obtain bounds for r.gl.dim K[S] in terms of r.gl.dim K [T _i ] and K where T _i  ⊂ S _i is a monoid. In addition, it is determined that K[S] has finite homological dimension if and only if each T _i has finite homological dimension.

2 Rees matrix semigroups

In the paper we use the notions and terminologies of Howie (Howie. J.M. 1976) and Weibel.

Assume that S is a monoid which has identity 1, zero element 0 and group of units U (S). Assume that Λ, I are sets which are not empty and P is a Λ × I matrix over S with entries p _λi where (λ, i) ∈ Λ × I. Then semigroup M = X (S ; I ; Λ ; P) is the set of triples I × S × Λ with a adjoint zero element. The elements in the form of (i, 0, λ) are also described as 0. A multiplication in M is defined as follows: ( i , x , λ ) ( j , y , μ ) = { ( i , xp λ j y , μ ) if p λ j ≠ 0 , 0 otherwise .

The matrix P is named regular if every column and every row have an element from U (S). All of these type semigroups can be supposed to be over a monoid and to have regular sandwich matrices. The set of idempotents is described by E or E (M). Let P ( M ) = { ( i , x , λ ) : x = p λ i - 1 , p λ i ∈ U ( S ) } ∪ { 0 } .

A monoid which is with a unique idempotent is called unipotent monoid.

Relations and are defined as follows: ( i , x , λ ) L ˜ ( j , y , μ ) if and only if ( for every e ∈ P ) , ( i , x , λ ) e = ( i , x , λ ) ⇔ ( j , y , μ ) e = ( j , y , μ )

( i , x , λ ) R ˜ ( j , y , μ ) if and only if ( for every e ∈ P ) , e ( i , x , λ ) ⇔ e ( j , y , μ ) = ( j , y , μ )

Obviously, L ˜ and R ˜ is equivalence relation on S. Moreover, we have the two equivalences H ˜ = L ˜ ∧ R ˜ and D ˜ = L ˜ ∨ R ˜ .

The following lemmas give the properties of Rees matrix semigroups.

Lemma 1.1. (Lawson. M. 1990) If (i, x, λ), (j, y, μ) ∈ S, then ( 1 ) ( i , x , λ ) L ˜ ( j , y , μ ) if and only if ( i , x , λ ) = ( j , y , μ ) = 0 or both elements are nonzero and λ = μ. or both elements are nonzero and i = j. ( 2 ) ( i , x , λ ) R ˜ ( j , y , μ ) if and only if ( i , x , λ ) = ( j , y , μ ) = 0 ( 3 ) If ( a , b ) ∈ L ˜ , y ∈ M , and ay , by ≠ 0 , then ( ay , by ) ∈ L ˜ . ( 4 ) If ( a , b ) ∈ R ˜ , y ∈ M , and ya , yb ≠ 0 , then ( ya , yb ) ∈ R ˜ .

Lemma 1.2. (Lawson. M. 1990)

All the elements of P (S) ∖ {0} are D-related.

Lemma 1.3. Let M = M⁰ (S ; I ; Λ ; P), where S is a monoid with zero and P is a Λ × I-matrix in order that every column and every row contains at least one invertible element from S, containing at least one L ˜ -class and at least one R ˜ -class. Then for every L ˜ of M, there is a R ˜ satisfying:

L ˜ R ˜ = M .

Proof. (i) Since M = M⁰ (S ; I ; Λ ; P), then M ∖ { 0 } = ∪ { H ˜ i λ : ( i , λ ) ∈ I × Λ } , where H ˜ i λ is a nonzero H ˜ -class of M. Assume L ˜ is a L ˜ -class of M. Since P is regular, choose a R ˜ -class R ˜ such that H ˜ which contains an idempotent e coincidence to the identity of M. By the proof of Theorem 3.6 in (Lawson. M. 1990), there exists left or right multiplication by regular elements from H ˜ e to other H ˜ -classes, Hence L ˜ R ˜ = M .

We have now chosen R ˜ such that L ˜ R ˜ = M . Notice that R ˜ L ˜ ⊆ R ˜ ∩ L ˜ . Let a ∈ R ˜ ∩ L ˜ . By the hypothesis, since M is Rees matrix semigroup, there exists an idempotent e of M in order that e R ˜ a. Since e ∈ R ˜ , a ∈ L ˜ so that a = ea ∈ R ˜ L ˜ . Hence R ˜ L ˜ = R ˜ ∩ L ˜ , that is R ˜ L ˜ is a H ˜ -class of M with identity e. So R ˜ L ˜ is a monoid with zero.□

Lemma 1.4. (Weibel. C.A. 1994) Let f : M → R be a ring map, A be an R-module. Then as an M-module pd M ( A ) ≤ pd R ( A ) + pd M ( R ) equality holds for every A iff it holds for every A such that pd _R  (A) ≤1.

Lemma 1.5. (Nico. W.R.1971) Assume that S is a finite semigroup with identity and zero, K is any field, l.gl.dim K[S] = l.gl.dim K₀ [S].

Let {S _α  : α ∈ A} be a set of semigroups which have the same zero element 0 in order that S _α  ∩ S _β  = {0}. Let S = ∪ _α∈AS _α . In the following, we give a multiplication on S: for each x ∈ S _α , y ∈ S _β , xy = { xy , α = β , 0 , α ≠ β .

Therefore S is a semigroup and S _α S _β  = {0}  if α ≠ β. Clearly S is the 0-direct union of the semigroups S _α , α ∈ A.

3 Algebras of Rees matrix semigroup

Let S be a monoid. K[S] is the semigroup algebra. If S has a zero m, then the contracted semigroup algebra is K₀ [S] = K[S]/K[m] and if L is an ideal of S, then K₀ [S/I] = K [S]/K [L].

Assume that M = M⁰ (S ; I ; Λ ; P) is a Rees matrix semigroup, the algebra K[M] can be characterized in the form of matrix type algebras.

Let P be a Λ × I matrix over K. Let M be the set of all I × Λ matrices over K. Then M is a ring, and a multiplication which is given similarly to that of a Rees matrix semigroup; namely AC = A ∘ P ∘ C A , C ∈ M where ∘ is the matrix multiplication. Therefore M is the matrix type algebra with sandwich matrix P and is in the form of M = M (K ; I ; Λ ; P).

Lemma 2.1. Assume that M = M⁰ (S ; I ; Λ ; P) is a Rees matrix semigroup over a monoid S. Then the semigroup algebra K [M⁰ (S ; I ; Λ ; P)] is isomorphic to the matrix type algebra M (K [S] ; I, Λ ; P) of the semigroup algebra K[S].

Proof. First of all, define a mapping ϑ : M 0 ( S ; I , Λ ; P ) → M ( K [ S ] ; I , Λ ; P ) , ϑ ( ( i , x , λ ) ) → ( a jn )

Where a _jn  = x if j = i, n = λ and in other case a _jn  = 0. It is clear to show that ϑ is a homomorphism into the semigroup of M (K [S] ; I, Λ ; P). Since the images of the elements of M⁰ (S ; I ; Λ ; P) which are nonzero are independent, therefore the extension of ϑ to a homomorphism of K₀ [M⁰ (S ; I, Λ ; P)] → M (K [S] ; I, Λ ; P) is an isomorphism.□

Proposition 2.2. If M = M⁰ (S ; I ; Λ ; P). Then we have

The algebra K₀ [M] has a identity.

Proof. Assume that K₀ [M] has a identity element E. Then XE = X ∘ P ∘ E = X EY = E ∘ P ∘ Y = Y for every X, Y ∈ M. Hence I = Λ. For if I > Λ, there is a nonzero Λ × I matrix over S such that X ∘ P = 0, which contradicts X ∘ P ∘ E = X ≠ 0. Similarly, Λ > I leads to a contradiction with E ∘ P ∘ Y = Y. It is that P is neither a left nor right divisor of zero in M_|I| (K [S]). Hence M_|I| (K [S]) has an identity and P is invertible.□

Conversely, assume that I, Λ are finite sets which have the same cardinality, Let P be an invertible matrix in M_|I| (K [S]). If P^-1 is an inverse of P in M_|I| (K [S]), E = P^-1 is the identity element in Rees matrix semigroup M.

Following the above lemma, we can have

Proposition 2.3. If S is a monoid with an ideal M satisfied that M is isomorphic to a semigroup which is in the form of M (T ; I, Λ ; P) where T is a monoid having zero and P is a Λ × I-matrix in order that every row and every column contains at least one invertible element from T. Let U = K[M]. Then L is an ideal of K₀ [S] that is satisfied the following properties:

L has an idempotent e satisfied that U = UeU = K₀ [S] eK₀ [S] and eU = eK₀ [S].

4 The homological dimension of some Rees matrix semigroups algebra

Lemma 3.1. Asuume that S is a semigroup, M = M (T ; I, Λ ; P) is an ideal of a S, where T is a monoid which has zero and P is a Λ × I-matrix satisfied that every row and every column contains at least one invertible element from T. Then max {r.gl.dim eK₀ [S] e, r.gl.dim K₀ [S/M]}≤ r.gl.dim (K₀ [S]) ≤ max {r . gl . dimeK₀ [S] e, r.gl.dim K₀ [S/M] +1} +1.

Proof. Since M = M (T ; I, Λ ; P) is a ideal of S, U = K [M] = M (K [T] ; I, Λ ; P) is an ideal of K[S]. Using Theorem 2.4., U contains an idempotent e such that U = UeU = K₀ [S] eK₀ [S], U_K0[S] is projective, and ek₀ [S] e ^U is projective. Also by Lemma Theorem 2.3. (5), we have that eK₀ [S] e ≅ K [T].□

By Proposition 1.2 of (Shapiro. J. 1993), r.gl.dim K = r.gl.dim eK₀ [S] e ≤ r . gl . dim eK 0 [ S ] e + pd ( Ue eK 0 [ S ] e ) + fd ( eK 0 [ S ] e eU ) = r . gl . dim K 0 [ S ]

Assume that M is a right K₀ [S/M]-module with pd (M_K0[S/M])< ∞. Then by Lemma 1.4,

pd (M_K0[S])≤ pd (M_K0[S/M]) +1 < ∞. Apply M ⊗ _K0[S] to the following short exact sequence 0 → U → K 0 [ S ] → K 0 [ S / M ] → 0 to have the following long exact sequence 0 → Tor 1 K 0 [ S ] ( M , K 0 [ S / M ] ) → M ⊗ K 0 [ S ] U → M ⊗ K 0 [ S ] K 0 [ S ] → M ⊗ K 0 [ S ] K 0 [ S / M ] → 0

Since U² = U and since MU = 0, we have M ⊗ _K0[S]U = 0 and hence Tor 1 K 0 [ S ] (M, K₀ [S/M]) =0 .

Since pd (K₀ [S/M]) _K0[S] ≤ 1, Tor p K 0 [ S ] ( M , K 0 [S/M]) =0 for all p > 0. If we apply ⊗_K0[S] (K₀ [S/M]) to a K₀ [S]-projective resolution of the semigroup M, we get a K₀ [S/M]-projective resolution of M; hence pd (M_{K0[S/M])≤pd(MK0[S]}). Thus r.gl.dim K₀ [S/M]≤r.gl.dim K₀ [S].

For the other side, if f = 1-e, by Theorem 1.4 of (Shapiro. J. 1993) we can have that r . gl . dim K 0 [ S ] ≤ max { r . gl . dim eK 0 [ S ] e + 1 , r . gl . dim fK 0 [ S ] f / fUf + pd ( K 0 [ S / M ] k 0 [ s ] ) + 1 } .

The result follows since fK 0 [ S ] f / fUf ≅ K 0 [ S ] / U If S = ∪ i = 1 n S i be the 0-direct union of Rees matrix semigroups S _i which is in the forms of S _i  = M (T _i  ; I _i , Λ _i  ; P _i ), then S = I 0 ⊇ I 1 = ∪ k = 1 n - 1 S k ⊇ … ⊇ I n = S n is an ideal chain of S.

Theorem 3.2. If S is a monoid which is the 0-direct union of finite Rees matrix semigroups S _i  = (T _i  ; I _i , Λ _i  ; P _i ) and if K is a commutative ring, then:

r.gl.dim K ≤ r.gl.dim K₀ [S]≤ r.gl.dim K+2n-2.

Proof. We apply the induction method on n.□

Assume n = 1, then S = (T ; I, Λ ; P). Since T is a monoid which has an identity, and in this case I = Λ = 1, Hence S is a semigroup with identity. By Lemma 3.1, we can obtain r . gl . dim eK 0 [ S ] e ≤ r . gl . dim ( K 0 [ S ] ) ≤ r . gl . dim eK 0 [ S ] e + 1

Hence r . gl . dim  (K₀ [S]) = r . gl . dim eK₀ [S] e = r . gl . K. Inductively, assume that the result is right for every monoids which is a union of less than n of finite Rees matrix semigroups. If U = K₀ [S _n ], then U is an ideal of K₀ [S] and by Theorem 2.3.(5), U has an idempotent e such that U = UeU = K₀ [S] eK₀ [S], U_K0[S] is projective, and _eK0 [S] eU is projective. Also by Theorem 2.3. (5), we have that eK₀ [S] ≅ K [T _n ]. By Lemma 3.1, r . gl . dim eK₀ [S] e = r . gl . dim K. Since K₀ [S]/U ≅ K₀ [S/S _n ] , S/S _n has a series of ideals with Rees matrix semigroup factors and by hypothesis the length of the sequence is less than n, the induction hypothesis gives that r . gl . dim K 0 [ S ] / U ≤ r . gl . dim K + 2 ( n - 1 ) - 2 = r . gl . dim K + 2 n - 4

Consequently, Lemma 3.1 implies that r . gl . dim K 0 [ S ] ≤ r . gl . dim K + 2 n - 4 + 2 = r . gl . dim K + 2 n - 2

To see the other inequality, observe that again by Lemma 3.1, r . gl . dim  K₀ [S] ≥ r . gl . dim  eK₀ [S] e = r . gl . dim K. The same method gives a proof of (2), from Lemma 3.1, K₀ [S] has finite global dimension if both eK₀ [S] e and K₀ [S/S _n ] do.

Conflict of interest

The author states that there is no conflict of interest in the publication of the paper.