On 0-homology of categorical at zero semigroups

On 0-homology of categorical at zero semigroups B. V. No viko v, L. Y u. P oly ako v a ∗ Abstract. The isomorphis m of 0-homology groups of a categorica l at z ero s emigroup and homo logy groups of its 0-reflector is pr ov ed. Some applications o f 0-homolog y to Eilenberg– MacLane homology o f semig roups ar e g iven. Key w ords. Homology o f semigroups, 0 -homolog y of semig r oups, catego rical at zero semigroup. Homology of semigroups and monoids w as defined in w orks b y Eilenberg and MacLane, but it w as not dev elop ed w ell la ter o n a nd tur ned out to b e less in ve stigated than cohomology of semigroups. Nev ertheless they find their application for differen t problems. F or instance , it is w ell-kno wn [2] that if a group G is a group of fractions of its submonoid M then H n ( G, A ) ∼ = H n ( M , A ) for ev ery G - mo dule A . In suc h a situation Dehorno y and Lafont [4] construct free resolutions for mo no ids whic h allow , in particular, to compute the homology of braid groups. In [13] Squier show ed that ev ery monoid, p ossessing a finite complete rewriting system, satisfied some homological condition. He answ ered nega- tiv ely the question o n an existence o f suc h a system for ev ery finitely pre- sen ted monoid with the solv able w ord problem. Squier’s approac h w as de- v elop ed, for instance, in [7] where t he metho d to construct a free resolution for monoids with a complete rewriting system w as described, whic h allow ed in its turn to find homology of suc h monoids. Homology of free partia lly comm utat ive monoids arise in the articles b y Husaino v (see [6], [5]) in connec tion with the construction of homology groups of async hro nous transition systems. If a semigroup S contains the zero then its homology and cohomolog y are trivial. In [9] (see also [11]) so called 0-ho mo lo gy w a s built whic h, g enerally ∗ Suppo rted b y N. I. Akhiezer foundation grant. 1 sp eaking, is nontrivial fo r semigroups with zero. F urthermore, if S con tains the zero then a semigroup ¯ S , called 0-reflector o f S , can b e constructed (see definition b elo w) and its cohomolog y groups are closely connected with 0- cohomology gro ups of S . Moreo v er for categorical at zero semigroups these groups are isomorphic in all dimensions. In particular 0-coho mo lo gy a llo ws to compute cohomology g roups in some cases. In view of the aforesaid in [12] 0 -homology of semigroups is constructed and it is shown t ha t the prop erties of the first 0-homolog y groups are similar to those of the first 0 -cohomology groups. In this w ork w e study 0-homolog y groups of greater dimensions. The main result ab out an isomorphism of 0- homology groups of the catego rical at zero semigroup a nd homology groups of its 0-reflector is contained in Section 2 (notice that its pro of essen tially differs from the pro of of the similar statement f or 0-cohomology). Section 3 is dev oted to the defining relations of categorical at zero semigroups and is auxiliary (how ev er is of its own interes t). It is used in Section 4 in examples and applications of 0-homology to t he computation o f Eilen b erg–MacLane homology groups. 1 Preliminaries and basic defi n itions All the mo dules under consideration are right mo dules. The notion Sem is used f o r the category of semigroups. Consider Sem 0 — the category , whic h ob jects are semigroups with zero elemen ts, and morphisms are suc h mappings ϕ : S → T that ϕ (0) = 0, ϕ − 1 (0) = 0 and ϕ ( xy ) = ϕ ( x ) ϕ ( y ) if xy 6 = 0 ( 0 -homomorphism s ). The sub category of the category Sem 0 consisting o f semigroups with an adjoint zero is, ob viously , isomorphic to the category Sem . Therefore w e will consider Sem as the sub catego ry in Sem 0 . Recall the definition of a reflectiv e sub category . Definition 1.1 [8] A s ub c ate gory D of a c ate gory C is c al le d r efle ctive if to e ac h obje ct C ∈ C such an obje ct R D ( C ) ∈ D (c al le d D -r e fle ctor of C ) and a m orphism ε D ( C ) : C → R D ( C ) ar e assigne d that for every D ∈ D the diagr am C ε D ( C ) − → R D ( C ) ↓ D 2 c an b e uniquely c omp l e te d by a morphism fr om Ho m D ( R D ( C ) , D ) up to c om- mutative one. In [12] it w as shown that the category Sem is reflectiv e in Sem 0 . F or a semigroup S with zero w e denote its Sem - reflector b y ¯ S and call it a 0- reflector. F or instance , the sem igroup S with an adjoint zero has a trivial 0-reflector: ¯ S = S . The semigroup ¯ S admits o ther equiv alent constructions. Let S b e giv en b y nonzero g enerato r s and defining relations: S = h a 1 , . . . , a k | P i = Q i , 1 ≤ i ≤ n i . (1) W e say tha t a relation P i = Q i is a zer o r elation if the v alue o f t he w ord P i in the semigroup S equals 0. Prop osition 1.2 [10, 12] . If al l the zer o r elations in (1) ar e thr own off then the semigr oup obtain e d i s the 0-r efle ctor of S . The following construction [9, 10] is con ven ien t for direct work with the elemen ts of the semigroup ¯ S . Let ( S ) denote the set of all sequences ( s 1 , s 2 , . . . , s n ), n ≥ 1, for whic h the fo llowing conditions hold: s i ∈ S \ 0 for ev ery 1 ≤ i ≤ n ; s i s i +1 = 0 for eve ry 1 ≤ i ≤ n − 1 . Define on the set ( S ) suc h a binary relation ν that ( s 1 , . . . , s m ) ν ( t 1 , . . . , t n ) if and only if one o f the follo wing conditions holds: 1) m = n and there exists i (1 ≤ i ≤ m − 1) suc h t hat s i = t i u, t i +1 = us i +1 for some u ∈ S and s j = t j if j 6 = i , j 6 = i + 1 ; 2) m = n + 1 and there exists i (2 ≤ i ≤ m − 1) suc h that s i = uv , t i − 1 = s i − 1 u , t i = v s i +1 for some u, v ∈ S , a nd s j = t j if 1 ≤ j ≤ i − 2, a nd s j = t j − 1 if i + 2 ≤ j ≤ m . Let ¯ ν b e the least equiv alence con taining ν and ¯ S b e a quotien t set ( S ) / ¯ ν . Let h s 1 , . . . , s n i denote the image of the elemen t ( s 1 , . . . , s n ) ∈ ( S ) under factorization. Then ¯ S b ecomes a semigroup whic h elemen t s are m ultiplied b y the fo llo wing rule: h s 1 , . . . , s m ih t 1 , . . . , t n i =  h s 1 , . . . , s m − 1 , s m t 1 , t 2 , . . . , t n i if s m t 1 6 = 0 ; h s 1 , . . . , s m − 1 , s m , t 1 , t 2 , . . . , t n i if s m t 1 = 0 . 3 The following notions giv e us an Abelian category r equired for building 0-homology . Definition 1.3 [9] A 0 -mo dule over a se m igr o up S with zer o is an A b elian gr oup A (in add itive notation) with a multiplic ation A × ( S \ 0 ) → A satisfying for every s, t ∈ S \ 0 , a, b ∈ A the c ond itions 1) ( a + b ) s = as + bs , 2) if st 6 = 0 then ( as ) t = a ( st ) . A hom o morphism fr om a 0-mo dule A to a 0-mo d ule B (over S ) is an A b eli a n gr oups hom omorphism ϕ : A → B such that ϕ ( as ) = ϕ ( a ) s for every s ∈ S \ 0 , a ∈ A . 0-Mo dules ov er the semigroup S form the catego r y C 0 ( S ) that is isomor- phic to the category C ( ¯ S ) of ordinary mo dules ov er ¯ S [9]. The corresp ondence b et wee n ob jects of these categories is sp ecified in suc h a w ay . If A ∈ C 0 ( S ) then A b ecomes an ¯ S -mo dule by putting a h s 1 , . . . , s n i = ( . . . ( as 1 ) s 2 . . . ) s n for a ∈ A . If ¯ A ∈ C ( ¯ S ) then ¯ A can be transformed into a 0- mo dule o v er S b y putting ¯ as = ¯ a h s i for ¯ a ∈ ¯ A and s ∈ S \ 0. Define no w 0-homolo g y groups for a semigroup S with zero 0 [12]. Let A b e a 0-mo dule o ver S . By D n w e denote the subset of all n -t uples [ s 1 , . . . , s n ], where s j ∈ S , j = 1 , . . . , n , suc h that s 1 s 2 . . . s n 6 = 0 . Let C 0 n ( S, A ) ( n ≥ 1) de- note the set of all ( finite) linear com binations of elemen ts f r o m D n with co effi- cien ts in A . W e write down suc h a linear combination as P a s 1 ,...,s n [ s 1 , . . . , s n ] and call it a n n -dimensional 0-chain. W e put C 0 0 ( S, A ) = A . The sets C 0 n ( S, A ) ( n ≥ 0) are Ab elian gr o ups with resp ect to addition. Define b oundary homomorphisms ∂ n : C 0 n ( S, A ) → C 0 n − 1 ( S, A ) o n the gener- ators in an usual w a y: ∂ n ( a [ s 1 , . . . , s n ]) = as 1 [ s 2 , . . . , s n ] + n − 1 X i =1 ( − 1) i a [ s 1 , . . . , s i s i +1 , . . . , s n ]+ ( − 1) n a [ s 1 , . . . , s n − 1 ] , if n ≥ 2; ∂ 1 ( a [ s ]) = as − a. It is easy to see that ∂ n is we ll defined and is a b oundary homomorphism: ∂ n − 1 ∂ n = 0. Definition 1.4 Th e gr oup H 0 n ( S, A ) = K er ∂ n / Im ∂ n +1 , n ≥ 1 is c al le d an n -th 0-homol o gy gr oup of a se m igr o up S with c o effic i e nts in a 0-m o d ule A . 4 In o ther w ords t he 0-homology g r o ups H 0 n ( S, A ) are defined as the ho- mology groups of the complex C 0 ∗ : · · · ∂ 3 → C 0 2 ( S, A ) ∂ 2 → C 0 1 ( S, A ) ∂ 1 → A. Along with the 0 -homology of S we consider homology groups H n ( ¯ S , A ) of the 0-reflector ¯ S with coefficien ts in the mo dule A . According to one o f the definitions [2] they are the homolog y groups of the complex C ∗ : · · · δ 3 → C 2 ( ¯ S , A ) δ 2 → C 1 ( ¯ S , A ) δ 1 → A. Here C j ( ¯ S , A ) a re gro ups of c hains, i.e. linear com binatio ns of the fo rm X a x 1 ,...,x n [ x 1 , . . . , x n ] where a x 1 ,...,x n ∈ A , [ x 1 , . . . , x n ] are all the p ossible n -tuples o f ¯ S elemen t s and only finitely man y summands are nonzero. The boundar y homomorphisms δ n are similar to the 0 -c hains homomorphisms ∂ n . Remark. The zero homology group H 0 ( ¯ S , A ) equals A/ Ker δ 1 , Ker δ 1 b eing generated by all differences of the f orm at − a where a ∈ A , t ∈ ¯ S . Therefore it is nat ura l t o define a zero 0-homology gr o up as H 0 0 ( S, A ) = A/ Ker ∂ 1 . The g r o up Ker ∂ 1 is a subgroup in A generated b y all differences of the form as − a where a ∈ A , s ∈ S \ 0 . The equalit y as − a = a h s i − a induces the em b edding Ker ∂ 1 ֒ → Ker δ 1 . Since eac h generator a h s 1 , . . . , s m i − a of the group Ker δ 1 can b e represen ted as a h s 1 , . . . , s m i − a = ( . . . ( as 1 ) . . . ) s n − a =  ( . . . ( as 1 ) . . . s n − 1 ) s n − ( . . . ( as 1 ) . . . ) s n − 1  +  ( . . . ( as 1 ) . . . s n − 2 ) s n − 1 − ( . . . ( as 1 ) . . . ) s n − 2  + · · · +  ( as 1 ) s 2 − as 1  +  as 1 − a  , this embedding is surjectiv e. Hence, Ker ∂ 1 = K er δ 1 and H 0 ( ¯ S , A ) = H 0 0 ( S, A ) . 5 Notice that if S is a semigroup with the adjoin t zero then H 0 n ( S, A ) ∼ = H n ( S \ 0 , A ). If w e consider A as a 0- mo dule ov er the semigroup S and as an or dina r y mo dule ov er ¯ S then t he group homomor phism ε n : C 0 n ( S, A ) → C n ( ¯ S , A ) ( n ≥ 1), defined as ε n ( a [ s 1 , . . . , s n ]) = a [ h s 1 i , . . . , h s n i ] , arises in a natural w a y . W e put also ε 0 = id A . The homomorphisms family ε = { ε n } ∞ n =0 can b e represen ted as a complex map: . . . ✲ C 0 2 ( S, A ) ✲ C 0 1 ( S, A ) ✲ A . . . ✲ C 2 ( ¯ S , A ) ✲ C 1 ( ¯ S , A ) ✲ A ❄ ❄ ❄ ε 2 ε 1 ε 0 ∂ 3 ∂ 2 ∂ 1 δ 3 δ 2 δ 1 It is not difficult to c heck that for i ≥ 1 the equalities ε i − 1 ∂ i = δ i ε i hold. Hence, the map ε = { ε n } ∞ n =0 is a c ha in map. Th us the homomor phisms ε n induce the homomorphisms ε ∗ n : H 0 n ( S, A ) → H n ( ¯ S , A ). F or an arbitrary semigroup S with zero and a 0-mo dule A the follo wing result was obta ined in [12] (taking in to accoun t the remark, giv en ab ov e, ab out the isomorphism of zero homolo gy groups): Theorem 1.5 ε ∗ k is an isomorphism for k ≤ 1 and an epimorp h ism for k = 2 . 2 The main theorem In what follows A is a fixed 0 -mo dule ov er a semigroup S unless sp ecified otherwise. Definition 2.1 [3] A semig r oup S is c al le d c a te goric al at zer o if xy z = 0 implies xy = 0 or y z = 0 . Our main result is contained in the following theorem: Theorem 2.2 I f S is c ate go ric al at zer o then the map ε ∗ n : H 0 n ( S, A ) → H n ( ¯ S , A ) is an isomorphism for al l n ≥ 0 and eve ry 0-mo dule A . 6 In view o f Theorem 1.5 the statemen t has to b e pro v ed only for n ≥ 2. T o pro ve that ε ∗ n is a monomorphism w e mak e use of the follo wing Lemma: Lemma 2.3 [12 ] L et chain c omplexes M , N and a chain ma p α : M → N b e given: . . . . . . ✲ M k +1 ✲ M k ✲ M k − 1 ✲ . . . . . . ✲ N k +1 ✲ N k ✲ N k − 1 ✲ ❄ ❄ ❄ ∂ k +1 ∂ k δ k +1 δ k α k +1 α k α k − 1 If for some k ≥ 1 ther e e xist mo dule homomorphisms β j : N j → M j ( j = k , k + 1 ) such that β k α k − id M k = 0 , (2) ∂ k +1 β k +1 = β k δ k +1 , (3) then the induc e d hom o lo gy gr oups homomorphis m α ∗ k : H k ( M ) → H k ( N ) is a monomorphi s m . W e put M k = C 0 k , N k = C k , α k = ε k and construct suitable homomor- phisms β k . The following notations will b e con v enien t: let X i l i denote an elemen t h x i 1 , . . . , x i l i i ∈ ¯ S . Besides if for a n n -dimensional c hain a [ X 1 l 1 , . . . , X n l n ] ∈ C n ( ¯ S , A ) the conditions l j 6 = 1, l j +1 = · · · = l n − 1 = 1 hold, we put x j = x j l j , x i = x i 1 , i = j + 1 , . . . , n . Define ho mo mo r phisms β n for n ≥ 2 on the g enerato r s of groups C n ( ¯ S , A ) β n ( a [ X 1 l 1 , . . . , X n l n ]) =    aX 1 l 1 − 1 [ x 1 , x 2 , . . . , x n ] , if l 2 = · · · = l n − 1 = 1 and x 1 x 2 . . . x n 6 = 0 ; 0 , otherwise. and then extend them b y linearit y . In the pro ofs of the follo wing tw o lemmas it is sufficien t to verify iden t it ies o n the generators of corresp onding groups. That is what w e will use. Lemma 2.4 F or n ≥ 2 the e quality β n ε n = id C 0 n holds. Pro of. F or n ≥ 2 and a [ s 1 , . . . , s n ] ∈ C 0 n w e hav e: β n ε n ( a [ s 1 , . . . , s n ]) = β n ( a [ h s 1 i , . . . , h s n i ]) = a [ s 1 , . . . , s n ] , since s 1 s 2 . . . s n 6 = 0.  7 Lemma 2.5 L et S b e a c ate goric al at zer o semigr oup. Th en ∂ n β n = β n − 1 δ n for al l n ≥ 2 . Pro of. F or n = 2 Lemma 2.5 is a sp ecial case of L emma 2.5 from [12]. Let n ≥ 3. Consider three p ossible cases for a generator a [ X 1 l 1 , . . . , X n l n ] ∈ C n ( ¯ S , A ). 1. Let l 2 = · · · = l n − 1 = 1 and x 1 x 2 . . . x n 6 = 0 . Then ∂ n β n  a [ X 1 l 1 , h x 2 i , . . . , h x n − 1 i , X n l n ]  = ∂ n  aX 1 l 1 − 1 [ x 1 , . . . , x n ]  = aX 1 l 1 [ x 2 , . . . , x n ] − aX 1 l 1 − 1 [ x 1 x 2 , . . . , x n ]+ n − 1 X j =2 ( − 1) j aX 1 l 1 − 1 [ x 1 , . . . , x j x j +1 , . . . , x n ] + ( − 1) n aX 1 l 1 − 1 [ x 1 , . . . , x n − 1 ] = β n − 1  aX 1 l 1 [ h x 2 i , . . . , h x n − 1 i , X n l n ] − a [ h x 1 1 , . . . , x 1 l 1 − 1 , x 1 x 2 i , h x 3 i , . . . , h x n − 1 i , X n l n ]+ n − 1 X j =2 ( − 1) j a [ X 1 l 1 , h x 2 i , . . . , h x j x j +1 i , . . . , h x n − 1 i , X n l n ]+ ( − 1) n a [ X 1 l 1 , h x 2 i , . . . , h x n − 1 i ]  = β n − 1 δ n  a [ X 1 l 1 , h x 2 i , . . . , h x n − 1 i , X n l n ]  . 2. Let l 2 = · · · = l n − 1 = 1 and x 1 x 2 . . . x n = 0 . Then ∂ n β n  a [ X 1 l 1 , h x 2 i , . . . , h x n − 1 i , X n l n ]  = 0 and β n − 1 δ n  a [ X 1 l 1 , h x 2 i , . . . , h x n − 1 i , X n l n ]  = = β n − 1  aX 1 l 1 [ h x 2 i , . . . , h x n − 1 i , X n l n ] − a [ X 1 l 1 h x 2 i , h x 3 i , . . . , h x n − 1 i , X n l n ]+ + n − 2 X j =2 ( − 1) j a [ X 1 l 1 , h x 2 i , . . . , h x j ih x j +1 i , . . . , h x n − 1 i , X n l n ]+ + ( − 1) n − 1 a [ X 1 l 1 , h x 2 i , . . . , h x n − 1 i X n l n ] + ( − 1) n a [ X 1 l 1 , h x 2 i , . . . , h x n − 1 i ]  . In this expression β n − 1 v anishes on all the summands of the in termediate sum. Notice that, if x 1 x 2 = 0 and x 2 x 3 . . . x n = 0 , then β n − 1 v anishes o n the other summands to o. If x 1 x 2 = 0 but x 2 x 3 . . . x n 6 = 0 then β n − 1 v anishes on eac h summand of the last pair and β n − 1  aX 1 l 1 [ h x 2 i , . . . , h x n − 1 i , X n l n ] − a [ X 1 l 1 h x 2 i , h x 3 i , . . . , h x n − 1 i , X n l n ]  = 0 , 8 since X 1 l 1 h x 2 i = h x 1 1 , . . . , x 1 l 1 − 1 , x 1 , x 2 i . If x 1 x 2 6 = 0 then x 2 x 3 . . . x n = 0 since S is categorical at zero. There- fore β n − 1 v anishes on eac h summand of the first pair. If at the same time x 2 x 3 . . . x n − 1 6 = 0 then categoricity a t zero implies x n − 1 x n = 0. Hence, h x n − 1 i X l n = h x n − 1 , x n , x n 2 , . . . , x n l n i and β n − 1  ( − 1) n − 1 a [ X 1 l 1 , h x 2 i , ..., h x n − 1 i X n l n ] + ( − 1) n a [ X 1 l 1 , h x 2 i , ..., h x n − 1 i ]  = 0 . Th us in this case β n − 1 δ n = 0 . 3. Finally consider the case l m > 1 for some 2 ≤ m ≤ n − 1. Again w e ha ve ∂ n β n  a [ X 1 l 1 , . . . , X n l n ]  = 0 and β n − 1 δ n  a [ X 1 l 1 , . . . , X n l n ]  = = β n − 1  aX 1 l 1 [ X 2 l 2 , . . . , X n l n ] − a [ X 1 l 1 X 2 l 2 , . . . , X n l n ]+ + n − 2 X j =2 ( − 1) j a [ X 1 l 1 , . . . , X j l j X j +1 l j +1 , . . . , X n l n ]+ + ( − 1) n − 1 a [ X 1 l 1 , . . . , X n − 1 l n − 1 X n l n ] − a [ X 1 l 1 , . . . , X n − 1 l n − 1 ]  . In this expression β n − 1 v anishes on the summands of the intermediate sum indep enden tly on m . If 2 < m < n − 1 then β n − 1 maps to 0 other summands as we ll. If m = 2 then β n − 1 v anishes on each summand of the last pair. Also β n − 1 either equals 0 on the b oth summands o f the first pair or maps their sum to 0. Similarly for m = n − 1. The lemma is pro v ed.  Lemma 2.6 L et H 0 k ( S, A ) ∼ = H k ( ¯ S , A ) for a sem igr oup S w ith zer o, k ≥ 1 and an arbitr a ry 0-mo dule A . Then the ma p ε ∗ k +1 : H 0 k +1 ( S, A ) → H k +1 ( ¯ S , A ) is an ep imorphism. Pro of. Consider ¯ S -mo dule A as a quotient mo dule of a free ¯ S -mo dule F b y a submo dule B . Thus w e ha v e a comm utative diagram with exact lines: . . . ✲ H 0 k +1 ( S, F ) ✲ H 0 k +1 ( S, A ) ✲ H 0 k ( S, B ) ✲ H 0 k ( S, F ) ✲ H 0 k ( S, A ) ✲ . . . . . . ✲ H k +1 ( ¯ S , F ) ✲ H k +1 ( ¯ S , A ) ✲ H k ( ¯ S , B ) ✲ H k ( ¯ S , F ) ✲ H k ( ¯ S , A ) ✲ . . . ❄ ❄ ❄ ❄ ❄ ε ∗ k +1 ε ∗ k Since F is a free mo dule H k ( ¯ S , F ) = H k +1 ( ¯ S , F ) = 0 and so H k ( ¯ S , B ) ∼ = H k +1 ( ¯ S , A ). Under the conditions H 0 k ( S, B ) ∼ = H k ( ¯ S , B ) and H 0 k ( S, F ) ∼ = 9 H k ( ¯ S , F ) = 0. It follows from here that t he map H 0 k +1 ( S, A ) → H 0 k ( S, B ) is an epimorphism. Finally comm utativit y of the diagram implies that the map ε ∗ k +1 : H 0 k +1 ( S, A ) → H k +1 ( ¯ S , A ) is an epimorphism as w ell.  The pro o f of Theorem 2.2. Lemmas 2.4, 2.5 imply conditions (2), (3) in Lemma 2.3 for n ≥ 2 b eing satisfied. Hence, t he map ε ∗ n is a monomorphism. Applying success iv ely Lemma 2.6 for k = 2 , 3 , . . . w e o btain the required statemen t.  3 Defining relations of cate g orical at zero se- migroups Denote by S = h a 1 , . . . , a n | A i = B i , i = 1 , . . . , r i a semigroup with gen- erators a j (1 ≤ j ≤ n ) a nd defining relations A i = B i , i = 1 , . . . , r . Let S be cat ego rical at zero. If some defining relation of the semigroup S is of the form A = 0 then, in view of categoricity at zero, it is a consequence of some equality a i a j = 0. Therefore in what f ollo ws w e supp ose that on the set N = { 1 , 2 , . . . , n } a relation Γ is giv en suc h that ( i, j ) ∈ Γ ⇔ a i a j = 0 and w e write do wn a categorical at zero semigroup in the follo wing form: S = h a 1 , . . . , a n | a i a j = 0 for ( i, j ) ∈ Γ; A k = B k , k = 1 , . . . , m i , (4) where A k 6 = 0 and B k 6 = 0 for a ll k ≤ m . In tro duce the notations: Γ a i = { a j | ( j, i ) ∈ Γ } , a i Γ = { a j | ( i, j ) ∈ Γ } . Besides denote the length of a w or d A b y l ( A ); we suppose that in (4) l ( A k ) ≥ l ( B k ) a nd l ( A k ) ≥ 1 for all k ≤ m . Prop osition 3.1 L et a semigr oup S b e giv e n in the form (4). L et A k = p k A ′ k q k and B k = r k B ′ k s k . The wor ds A ′ k and B ′ k c an b e empty and if l ( B k ) = 1 we supp ose that B k = r k = s k (her e p k , q k , r k , s k ∈ { a 1 , . . . , a n } ). The semigr oup S is c ate goric al at zer o if and only if Γ p k = Γ r k and q k Γ = s k Γ for al l k ≤ m . Pro of. Let S be categorical at zero and, for instance, a ∈ Γ p k . The n A k = B k implies ar k B ′ k s k = 0. Since B ′ k 6 = 0, in view of categor icity , ar k = 0 . Hence, a ∈ Γ r k . Next ve rify the con v erse statemen t. Let Γ p k = Γ r k , q k Γ = s k Γ and X Y Z = 0 , X Y 6 = 0, Y Z 6 = 0. If the w o r d X Y Z contains a pro duct a i a j = 0 10 then either X Y = 0 or Y Z = 0 whic h is imp o ssible. Therefore t he transfor- mation o f the w ord X Y Z in t o zero is r ealized only by equalities A k = B k . Ho we v er, according to the condition, the pro ducts a i a j = 0 cannot appear during suc h a transformation. Hence, con trary to the a ssumption either X Y = 0 o r Y Z = 0.  Consider now a connection b etw een defining relations of semigroup S and those of its 0- reflector ¯ S . Prop osition 3.2 L et a c a te goric al at ze r o semigr oup S b e given by defining r ela tion s (4). Then ¯ S = h h a 1 i , . . . , h a n i | A k = B k , k = 1 , . . . , m i , wher e the wo r ds A k , B k ar e c ons ider e d in the alph a b et h a 1 i , . . . , h a n i . C o n - versely if ¯ S is given by r elations A k = B k ( k = 1 , . . . , m ) then ther e exists a subset Γ ⊆ N such that the semigr oup S c an b e giv e n in the form (4). Pro of. The first part o f the prop osition follows immediately from Prop osi- tion 1.2. Let now the semigroup ¯ S b e defined b y t he relat io ns A k = B k ( k = 1 , . . . , m ) and C = D b e a n equality in S . If C 6≡ 0, D 6≡ 0 then this equalit y holds in ¯ S as w ell. Hence it is a consequenc e of the relations A k = B k . If, for instance, C 6≡ 0, D ≡ 0 and C ≡ a i 1 . . . a i r then categoricit y at zero implies a i k a i k +1 = 0 for some k , i.e. the equalit y C = 0 follow s from a i a j = 0 , ( i, j ) ∈ Γ. The second par t o f the statemen t is pro v ed.  4 Some appli c ations The results of the previous section can b e used to establish connections b e- t we en ordinar y homology gr o ups and 0-ho mology ones. The follow ing asser- tion is a simple example: Prop osition 4.1 L et al l the definin g r elations of a se migr oup S b e of the form a i a j = 0 . Then H 0 n ( S, A ) = 0 fo r al l n > 1 a nd every 0-mo dule A over S . Pro of. According to Prop osition 3.1 S is categorical at zero. Prop osition 3.2 implies that ¯ S is a free semigroup. Hence, H n ( ¯ S , A ) = 0 for n > 1 (see, for example, [2]) . No w the statemen t follows from Theorem 2.2.  11 Usually a semigroup with zero is simpler t han its 0-reflector. Therefore for computation of homology gro ups of a giv en semigroup T the follo wing tec hnique can b e used: fin d a categorical at zero semigroup S suc h that its 0-reflector ¯ S is isomorphic to T , calculate H 0 n ( S, ) and use Theorem 2.2. Let a semigroup T b e giv en in the form T = h a 1 , . . . , a n | A k = B k , k = 1 , . . . , m i (5) In tro duce the nota tion: P = { A k = B k | 1 ≤ k ≤ m } . Let I ( P ) denote the set of the elemen ts x ∈ T suc h that A k 6∈ T xT for all k . This set is an ideal in T if it is not empt y . The f ollo wing prop o sition prov ed in [12] is in some sense the con v erse to Prop osition 1.2. It will b e helpful fo r us in examples. Prop osition 4.2 L et a semigr oup T b e give n in the form (5) and I ( P ) 6 = ∅ . If a j 6∈ I ( P ) for al l 1 ≤ j ≤ m then T is a 0- r efle ctor of the quotient semigr oup S = T /I ( P ) . Example 1. Consider the semigroup T = h a, b, c, d | ab = cd i . Then T \ I ( P ) = { a, b, c, d , x = ab = cd } and by Prop o sition 4.2 T = ¯ S where S consists o f the elemen ts 0 , a, b, c, d , x , all the pro ducts b eing equal zero except ab = cd = x . Since S 3 = 0 w e ha ve H 0 2 ( S, A ) = Ker ∂ 2 for eac h 0-mo dule A o ver S . An arbitrary 2-dimensional 0 -cycle is of the form f = α [ a, b ] + β [ c, d ]. The equality ∂ 2 f = 0 implies α = β = 0 and so H 0 2 ( S, A ) = 0. Hence H 2 ( T , A ) = 0 b y Theorem 1.5. This implies H n ( T , A ) = 0 for ev ery n ≥ 2 and ev ery T -mo dule A . Example 2. Let T = h a, b, c | ab = ac i . In this example T \ I ( P ) = { a, b, c, ab } . Then by Prop osition 4.2 T = ¯ S where S consists of the elemen ts 0 , a, b, c, ab . Again S 3 = 0 and w e ha ve H 0 2 ( S, A ) = Ker ∂ 2 . Let f = α [ a, b ] + β [ a, c ] b e a 2- dimensional 0-cycle. Then t he equality ∂ 2 f = 0 implies αa = 0 and β = − α . Th us the gro up H 0 2 ( S, A ) is isomorphic to the subgroup A a of the 0-mo dule A consisting of the elemen ts α suc h that α a = 0. It is not difficult to v erify that S is a categorical at zero semigroup. So b y Theorem 2.2 H 2 ( T , A ) ∼ = A a . Let a semigroup T b e giv en in the form (5). Assign to the defining relations system P the gra ph ∆, whic h v ertices set { 1 , 2 , . . . , n } and the edges are the pairs ( i, j ) such that the pro duct a i a j is contained in some of the words A k , B k ( k ≤ m ). W e call a v ertex a of the graph ∆ an entr an c e ( an exit ) if ( b, a ) 6∈ ∆ (resp ectiv ely ( a, b ) 6∈ ∆) for ev ery v ertex b . 12 Theorem 4.3 L et a se migr oup T , given in the form (5), s a tisfy the fol lowing c on dition: for al l the w o r ds A k , B k ( k ≤ m ) their fi rst letters ar e entr anc es and the las t letters ar e exits in the gr aph ∆ . Then the s e migr oup S = h x 1 , . . . , x n | x i x j = 0 ⇔ ( i, j ) 6∈ ∆; A k = B k , k = 1 , . . . , m i , wher e the wor ds A k , B k ar e written down in the alphab et { x 1 , . . . , x n } , is c ate goric al at zer o and T is a 0-r efle ctor of S . Pro of. Let A k = p k A ′ k q k , B k = r k B ′ k s k where p k , q k , r k , s k ∈ { x 1 , . . . , x n } . Since p k and r k are entrances Γ p k = Γ r k = { x 1 , . . . , x n } and similarly q k Γ = s k Γ. Because of the same reason none of the w o rds A k , B k , ( k ≤ m ) contains the other. Therefore in the semigroup no ne of the defining relations A k = B k is a zero relation. According to Prop osition 3.1 S is categorical at zero. Prop osition 3.2 implies that its 0-reflector is isomorphic to T .  Corollary 4.4 L et a semi g r oup T is under the c ondi tions of the pr evio us the or em and the gr aph ∆ do e s no t c ontain any c ir cuit. T hen H l ( T , A ) = 0 for al l l > l 0 + 1 wher e l 0 is the length of the longest p ath in ∆ . Pro of. Consider the semigroup S from Theorem 4.3. In consequence of the absence of circuits ev ery w ord in S of the length greater than l 0 + 1 equals zero and so S l 0 +2 = 0. Therefore C l 0 ( S, A ) = 0 as so on as l > l 0 + 1. Hence, H l ( T , A ) ∼ = H 0 l ( S, A ) = 0.  Example 3. Consider the Ady an semigroup [1]: T = h a, b, c, d, e | ab = cd, aeb = ced i . The graph ∆ for it is of the form: ✟ ✟ ✟ ✟ ✯ ❍ ❍ ❍ ❍ ❥ ❍ ❍ ❍ ❍ ❥ ✲ ✟ ✟ ✟ ✟ ✯ ✲ a c b d e Therefore all homology groups o f dimension 4 and greater are trivial. In conclusion w e consider free pro ducts o f semigroups. In [12] t he description of the first homology gr oup of free pro duct of t wo semigroups was obtained. If S and T are semigroups without zero and A is a 13 S ∗ T -mo dule w e denote by A ( S − 1) (resp ectiv ely b y A ( T − 1)) a subgroup in the mo dule A generated b y the elemen ts of the form as − a where a ∈ A, s ∈ S (resp ectiv ely at − a where a ∈ A, t ∈ T ) and put A 1 = A ( S − 1) ∩ A ( T − 1). Then the following pro p osition holds: Prop osition 4.5 H 1 ( S ∗ T , A ) is an ex tens ion of H 1 ( S, A ) ⊕ H 1 ( T , A ) by A 1 . In particular, if A is a trivial S ∗ T -mo dule then H 1 ( S ∗ T , A ) ∼ = H 1 ( S, A ) ⊕ H 1 ( T , A ). No w pro ceed to the homology gro ups of greater dimensions. Recall [3] that a semigroup U is called a 0-direct union of semigroups { S λ } λ ∈ Λ if U = S λ ∈ Λ S λ , S λ ∩ S µ = 0 and S λ S µ = 0 for a ll λ 6 = µ . Lemma 4.6 L et a semigr oup S b e a 0-dir e ct union of sem igr oups S λ , ( λ ∈ Λ) . Then H 0 n ( S, A ) ∼ = M λ ∈ Λ H 0 n ( S λ , A ) , wher e A is a 0-mo dule over S (and so over every S λ as wel l) and n > 1 . Pro of. Let c = P a s 1 ...s n ⊗ ( s 1 , . . . , s n ) b e an n - dimensional 0-cycle from C 0 n ( S, A ). Then fo r ev ery summand a s 1 ...s n ⊗ ( s 1 , . . . , s n ) all elemen ts s j b e- long to the same semigroup S λ , otherwise c is not defined. The refore ev ery cycle can b e giv en as the sum c = Σ λ ∈ Λ c λ where c λ is a cyc le belonging to C 0 n ( S λ , A ). At the same time c = ∂ c ′ for some c ′ ∈ C 0 n +1 ( S, A ) if and only if c λ = ∂ c ′ λ for all λ ∈ Λ, what implies the statemen t.  Prop osition 4.7 L et S = Q ∗ λ ∈ Λ S λ b e a fr e e pr o duct of semigr oups S λ . Then H n ( S, A ) ∼ = M λ ∈ Λ H n ( S λ , A ) for every S -mo dule A an d e ach n > 1 . Pro of. Similarly to the pro of of Theorem 5 in [9 ] consider T — the 0-direct union of the semigroups T λ = S λ ∪ 0 with extra zero es. Then the semigroup T is categorical a t zero and T ≃ S . In view of Theorem 2.2 a nd the previous lemma we obtain t he required statemen t.  14 References [1] S. I. Ady an: “Defining relations and a lgorithmical problems fo r groups and semigroups”. T rudy matema t. instituta imeni V. A. Steklova , V ol. 85, (19 6 6) (in Russian). [2] H. Cartan and S. Eilen b erg: Homo l o gic al algebr a , Princeton, 1956. [3] A. H. Clifford and G. B. Preston: The algebr aic the ory of semigr oups. V ol. II , American Mathematical So ciety , 1 967. [4] P . D ehorno y and Yv. Laf on t: “Homology of Gaussian groups”, Ann. Inst. F ourier , V ol. 53(2), (2 003), pp. 489–540. [5] A. A. Husaino v: “On the homology of small categories and async hro nous transition systems”, Homolo g y, Homotopy and Appli c ations , V o l. 6 (1), (2004), pp. 439– 471. [6] A. A. Husaino v, V. V. 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