Essential variables and positions in terms

ESSENTIAL V ARIABLES AND POSITIONS IN TERMS SLA VCHO SHTRAKO V Abstract. The paper deals with Σ − composition of terms, which allows us to extend the deriv ation rules in f ormal deduction of i dentities. The concept of essen tial v ariables and essen tial p ositions of te rms with r espect to a set of iden tities i s a k ey step in the simplification of the pro cess of formal de- duction. Σ − composition of terms is defined as replacemen t betw een Σ- equal terms. This composition induces Σ R − deductive ly closed s ets of iden tities. In analogy to ba lanced identities we i n tro duce and inv estigate Σ − balanced iden tities for a giv en set of iden tities Σ. 1. Introduction Let F b e any finite s et, the elements of whic h are calle d operati on sy mbol s. Let τ : F → N be a mapping in to the non-negative int eger s; for f ∈ F , the n um b er τ ( f ) will denote th e arity of the op eration s y m b ol f . The pair ( F , τ ) is called a typ e or signatur e . If it is obvious what the s et F is , we will write “ ty pe τ ”. The set of symbols of arity p is denoted b y F p . Let X be a finite set of v ariables, and let τ b e a type with the set of op eration symbols F = ∪ j ≥ 0 F j . The set W τ ( X ) of terms of t yp e τ with v ariables from X is the smallest set such that (i) X ∪ F 0 ⊆ W τ ( X ); (ii) if f is an n − a ry op eration symbo l and t 1 , . . . , t n are terms, then the “string” f ( t 1 . . . t n ) is a term. An algebra A = h A ; F A i of t ype τ is a pair consisting of a s e t A and an indexed set F A of op era tions, defined o n A . If f ∈ F , then f A denotes a τ ( f )-ar y op eration on the set A. W e denote by A lg ( τ ) the cla ss of all algebras of t yp e τ . If s, t ∈ W τ ( X ), then the pair s ≈ t is called an identit y of type τ . I d ( τ ) denotes the set of all identities of t yp e τ . An ide ntit y t ≈ s ∈ I d ( τ ) is satisfied in the alg ebra A , if the term o p e r ations t A and s A , induced b y the terms t and s on the a lgebra A are equal, i.e., t A = s A . In this case we write A | = t ≈ s and if Σ is a set of identities of type τ , then A | = Σ means that A | = t ≈ s for all t ≈ s ∈ Σ. 2000 Mathema tics Subje ct Classific ation. Pri mary: 08B05; Secondary: 08A02, 03C05, 08B15. Key wor ds and phr ases. Co mp osition of terms, Essential position in a term, Globally inv ari- an t congruence, Stable v ariety . 1 2 SL. SHTRAKO V Let Σ b e a set of iden tities. F or t, s ∈ W τ ( X ) w e write Σ | = t ≈ s if, giv en any algebra A , A | = Σ ⇒ A | = t ≈ s. The o p er ators I d a nd M od a re defined for classes of algebra s K ⊆ A lg ( τ ) and for sets of identities Σ ⊆ I d ( τ ) a s follows I d ( K ) := { t ≈ s | A ∈ K ⇒ A | = t ≈ s } , and M od (Σ) := {A | t ≈ s ∈ Σ ⇒ A | = t ≈ s } . The set of fixed points with respect to the closure operator s I dM od and M odI d form c o mplete lattices L ( τ ) and E ( τ ) o f all v arieties o f type τ and of all equational theories (logics) of type τ . In [1] deductiv e closures of sets of identities are used to descr ibe some elements of these lattices. W e will apply the co ncept of Σ − co mp os itions of terms to study the lattices L ( τ ) a nd E ( τ ). W e use the concept of es s ent ial v ariables, a s defined in [5] and ther efore we co nsider such v ariables with resp ect to a given set of iden tities, which is a fully inv arian t congruence. In Section 2 w e inv estigate the co ncept of Σ − ess en tial v ariables and p ositio ns. The fictiv e (non-e ssential) v ariables and po sitions are used to simplify the deduc- tions of identities in equa tional theories. W e introduce Σ − comp os ition of terms for a given s e t Σ o f iden tities. In Sectio n 3 we de s crib e the closure op era to r Σ R in the set of all ident ities of a given type, whic h g enerate extensio ns o f fully inv a riant co ngruences. The v arieties which satisfy Σ R − closed sets a re fully inv ariant congr uences and they are c a lled stable. The stable v arieties ar e compared to solid ones [2, 4, 6]. In Section 4 we intro duce and study Σ − balance d iden tities and pro ve that Σ − balanced prop erty is clo sed under Σ R -deductions. 2. Compositions of terms If t is a term, then the set v ar ( t ) co nsisting of those elemen ts o f X which o ccur in t is called the set of input variables (or variables) for t . If t = f ( t 1 , . . . , t n ) is a non-v ariable term, then f is the r o ot symb ol (r o ot) of t and we will write f = r oot ( t ) . F o r a term t ∈ W τ ( X ) the set S ub ( t ) of its subterms is defined as follows: if t ∈ X ∪ F 0 , then S ub ( t ) = { t } and if t = f ( t 1 , . . . , t n ), then S ub ( t ) = { t } ∪ S u b ( t 1 ) ∪ . . . ∪ S ub ( t n ) . The dept h of a term t is defined inductively: if t ∈ X ∪ F 0 then D epth ( t ) = 0 ; and if t = f ( t 1 , . . . , t n ), then D epth ( t ) = max { D epth ( t 1 ) , . . . , D e p th ( t n ) } + 1 . Definition 2.1. Let r, s, t ∈ W τ ( X ) b e thre e terms of type τ . By t ( r ← s ) we will denote the term, o btained by sim ultaneous repla c emen t of e very occurr ence of r as a subterm of t b y s . This ter m is called the inductive c omp osition of the terms t and s , by r . In particula r, (i) t ( r ← s ) = t if r / ∈ S ub ( t ); (ii) t ( r ← s ) = s if t = r , a nd ESSENTIAL V ARIABLES AND POSITIONS IN T ERMS 3 (iii) t ( r ← s ) = f ( t 1 ( r ← s ) , . . . , t n ( r ← s )), if t = f ( t 1 , . . . , t n ) and r ∈ S u b ( t ), r 6 = t . If r i / ∈ S ub ( r j ) when i 6 = j , then t ( r 1 ← s 1 , . . . , r m ← s m ) means the inductiv e comp osition o f t, r 1 , . . . , r m by s 1 , . . . , s m . In the particular case when r j = x j for j = 1 , . . . , m and v ar ( t ) = { x 1 , . . . , x m } we will briefly write t ( s 1 , . . . , s m ) instead of t ( x 1 ← s 1 , . . . , x m ← s m ). An y term ca n b e reg arded as a tree with no des la belle d as the op era tion symbols and its leaves labelled as v ariable s o r nullary oper ation symbols. Often the tree of a term is prese nted b y a diagra m of the corres po nding term as it is sho wn b y Figure 1. Let τ be a type and F be its set of op eration symbo ls. Denote by maxar = max { τ ( f ) | f ∈ F } and N F := { m ∈ N | m ≤ maxar } . Let N ∗ F be the set of all finite strings ov e r N F . The set N ∗ F is naturally ordere d b y p  q ⇐ ⇒ p is a pr efix of q . The Gree k letter ε , as us ual denotes the empty word (string) ov er N F . T o distinguish b etw een different o ccur r ences of the same o p er ation symbol in a term t we assign to each op eration s ymbol a po sition, i.e., a n element of a g iven set. Usually positio ns a re finite se q uences (strings) of natural n umbers. Each p osition is assigned to a no de of the tree diagram of t , starting with the empt y sequence ε for the root and using th e in tegers j , 1 ≤ j ≤ n i for th e j -th branc h of an n i -ary op erational symbo l f i . So, le t the pos itio n p = a 1 a 2 . . . a s ∈ N ∗ F be assigne d to a no de of t lab elled by the n i -ary op erationa l symbo l f i . Then the position a ssigned to the j -th child of this no de is a 1 a 2 . . . a s j . The set of po sitions of a term t is denoted b y P os ( t ) and it is illustrated by Example 2.1. Thu s we hav e P os ( t ) ⊆ N ∗ F . Let t ∈ W τ ( X ) be a term of type τ and let s u b t : P os ( t ) → S ub ( t ) b e the function which maps each p osition in a term t to the subterm of t , whose ro o t no de o c c urs at that p osition. Definition 2.2. Let t, r ∈ W τ ( X ) be tw o ter ms of t yp e τ and p ∈ P os ( t ) b e a po sition in t. The pos itional compos itio n of t and r on p is the term s := t ( p ; r ) obtained from t by r eplacing the term sub t ( p ) by r on the p osition p , only . Example 2.1. Let τ = (2 ), t = f ( f ( x 1 , f ( f ( f ( x 1 , x 2 ) , x 2 ) , x 3 )) , x 4 ) and u = f ( x 4 , x 1 ). The pos itions o f t and u are written on their no des in Fig ur e 1. Then the positional compos ition o f t a nd u on the p osition 121 ∈ P os ( t ) is t (121; u ) = f ( f ( x 1 , f ( f ( x 4 , x 1 ) , x 3 )) , x 4 ) and sub t (121) = f ( f ( x 1 , x 2 ) , x 2 ). Remark 2.1. The po sitional comp osition has the following prop erties: 1. If hh p 1 , p 2 i , h t 1 , t 2 ii is a pair with p 1 6≺ p 2 & p 2 6≺ p 1 , then t ( p 1 , p 2 ; t 1 , t 2 ) = t ( p 1 ; t 1 )( p 2 ; t 2 ) = t ( p 2 ; t 2 )( p 1 ; t 1 ); 2. If S = h p 1 , . . . , p m i and T = h t 1 , . . . , t m i with ( ∀ p i , p j ∈ S ) ( i 6 = j ⇒ p i 6≺ p j & p j 6≺ p i ) 4 SL. SHTRAKO V    ✒    ✒    ✒    ✒    ✒    ✒    ✒    ✒ ❅ ❅ ❅ ■ ❅ ❅ ❅ ■ r r r r r r r r r r ❡ ❡ ❡ r r r r r 12 12 121 121 ε 1211 1211 1 1 1 x 1 x 1 11 11 122 122 1212 1212 2 12112 x 3 x 3 x 2 x 1 x 1 x 2    ✒    ✒ ❅ ❅ ❅ ■ ❅ ❅ ❅ ■ ❅ ❅ ❅ ■ ❅ ❅ ❅ ■ ❅ ❅ ❅ ■ ❅ ❅ ❅ ■ ❅ ❅ ❅ ■ ❅ ❅ ❅ ■ r r ε ε 2 2 x 4 x 4 12111 x 1 r r r x 4 x 4 t u t (121; u ) Figure 1. Positional comp osition of t erms and π is a per m utation o f the set { 1 , . . . , m } , then t ( p 1 , . . . , p m ; t 1 , . . . , t m ) = t ( p π (1) , . . . , p π ( m ) ; t π (1) , . . . , t π ( m ) ) . 3. If t, s , r ∈ W τ ( X ), p ∈ P os ( t ) and q ∈ P os ( s ), then t ( p ; s ( q ; r )) = t ( p ; s )( pq ; r ). 4. Let s, t ∈ W τ ( X ) and r ∈ S ub ( t ) be terms of t yp e τ . Let { p 1 , . . . , p m } = { p ∈ P os ( t ) | sub t ( p ) = r } . Then we hav e t ( p 1 , . . . , p m ; s ) := t ( p 1 ; s )( p 2 ; s ) , . . . , ( p m ; s ) = t ( r ← s ) , which shows that an y inductive comp osition can be represented as a p ositional one. O n the other side there are examples of p ositiona l compo sitions which can not be realized as inductiv e comp ositio ns. Definition 2.3. Let Σ ⊆ I d ( τ ), t ∈ W τ ( X n ) b e an n − ary term of t yp e τ , A = h A, F i be an algebra of t yp e τ and let x i ∈ v ar ( t ) b e a v ariable whic h occur s in t. (i) [5] The v ar iable x i is called essen t ial for t with resp ect to th e algebra A if there are n + 1 elemen ts a 1 , . . . , a i − 1 , a , b, a i +1 , . . . , a n ∈ A suc h that t A ( a 1 , . . . , a i − 1 , a , a i +1 , . . . , a n ) 6 = t A ( a 1 , . . . , a i − 1 , b , a i +1 , . . . , a n ) . ESSENTIAL V ARIABLES AND POSITIONS IN T ERMS 5 The set of all ess ent ial v ariables for t with respect to A will be denoted b y E ss ( t, A ). F i c ( t, A ) denotes the set of all v ariables in v a r ( t ), which are no t essen- tial with resp ect to A , called fictive ones. (ii) A v a riable x i is said to be Σ − essential for a ter m t if there is an algebr a A , suc h that A | = Σ and x i ∈ E ss ( t, A ) . The set o f all Σ − essential v ar iables for t will be denoted b y E ss ( t, Σ) . If a v ariable is not Σ − essential for t , then it is ca lled Σ − fictive for t . F ic ( t, Σ) denotes the set of all Σ − fic tive v ariables for t. Prop ositio n 2.1. If Σ 1 ⊆ Σ 2 ⊆ I d ( τ ) , t ∈ W τ ( X ) and x i ∈ E ss ( t, Σ 2 ) , then x i ∈ E ss ( t, Σ 1 ) . Theorem 2 .1. L et t ∈ W τ ( X ) a nd Σ ⊆ I d ( τ ) . A varia ble x i is Σ − essential for t if and only if ther e is a term r of typ e τ such that r 6 = x i and A 6| = t ≈ t ( x i ← r ) for some algebr a A ∈ A lg ( τ ) with A | = Σ . Pr o of. Let t ∈ W τ ( X n ) for some n ∈ N and let A ∈ A lg ( τ ) b e an algebr a for which A | = Σ and x i ∈ E ss ( t, A ). Then fr om Lemma 3.5 of [5] it follo ws that A 6| = t ≈ t ( x i ← x n +1 ) . Hence A 6| = t ≈ t ( x i ← r ) with r = x n +1 . Conv ersely , let us assume that there is a term r , r 6 = x i of type τ with A 6| = t ≈ t ( x i ← r ) for a n algebra A ∈ A l g ( τ ) with A | = Σ . Let m ∈ N be a natural n umber for whic h r ∈ W τ ( X m ). So , there are m + n v alues a 1 , . . . , a i − 1 , a i , a i +1 , . . . , a n , b 1 , . . . , b m ∈ A such that r A ( b 1 , . . . , b m ) 6 = a i and t A ( a 1 , . . . , a i − 1 , a i , a i +1 , . . . , a n ) 6 = t A ( a 1 , . . . , a i − 1 , r A ( b 1 , . . . , b m ) , a i +1 , . . . , a n ) . The last inequality sho ws that x i ∈ E ss ( t, A ) . Hence x i is Σ − esse n tial for t .  Corollary 2.1 . If t ≈ s ∈ Σ a nd x i ∈ F ic ( t, Σ) , then f or e ach term r ∈ W τ ( X ) , we h ave Σ | = t ( x i ← r ) ≈ s. Corollary 2.2. A variable x i is Σ − essential for t ∈ W τ ( X n ) if and only if x i is essential for t with r esp e ct to any M od (Σ) -fr e e algebr a with at le ast n + 1 fr e e gener ators. Corollary 2.3. L et Σ ⊆ I d ( τ ) b e a set of identities of t yp e τ and t ≈ s ∈ Σ . If a variable x i is Σ − fictive for t , then it is fictive fo r s with r esp e ct to e ach algebr a A ∈ M od (Σ) . The concept o f Σ − essential p os itions is a natural extension of Σ − essential v ari- ables. Definition 2.4. Let A = h A, F i b e an algebra of type τ , t ∈ W τ ( X n ), and let p ∈ P os ( t ). (i) If x n +1 ∈ E ss ( t ( p ; x n +1 ) , A ), then the po s ition p ∈ P os ( t ) is ca lled essential for t with resp ect to the algebra A . The set of all essential pos itions for t with resp ect to A is denoted by P E ss ( t, A ) . When a pos ition p ∈ P os ( t ) is not essential 6 SL. SHTRAKO V for t with resp ect to A , it is called fictive for t with respect to A . The s e t of all fictive p ositions with r esp ect to A is denoted by P F i c ( t, A ) . (ii) If x n +1 ∈ E ss ( t ( p ; x n +1 ) , Σ), then the p ositio n p ∈ P os ( t ) is called Σ − essential for t . The s et of Σ − essential positions for t is denoted by P E s s ( t, Σ) . When a p osition is not Σ − essential fo r t it is called Σ − fictive . P F ic ( t, Σ) denotes the set of all Σ − fictive positions for t. The set of Σ-ess e ntial subterms of t is defined as follows: S E ss ( t, Σ) := { sub t ( p ) | p ∈ P E ss ( t, Σ) } . S F ic ( t, Σ) denotes the set S F ic ( t, Σ ) := S ub ( t ) \ S E ss ( t, Σ). So, Σ-essential subterms of a term ar e subterms which o ccur at a Σ-ess ent ial po sition. Since one subter m can o ccur at more than o ne p osition in a term, and can o ccur in b oth Σ- essential and non-Σ-ess e n tial po sitions, we note that a subterm is Σ-e s sential if it o ccurs at least once in a Σ-essential p osition, a nd Σ - fictive otherwise. Example 2.2. Let τ = (2) and let t = f ( f ( x 1 , x 2 ) , f ( f ( x 1 , x 2 ) , x 3 )). L et us consider the v arie ty RB = M od (Σ) of recta ngular bands, where Σ = { f ( x 1 , f ( x 2 , x 3 )) ≈ f ( f ( x 1 , x 2 ) , x 3 ) ≈ f ( x 1 , x 3 ) , f ( x 1 , x 1 ) ≈ x 1 } . It is not difficult to see that t he Σ − essential p ositions and subterms of t a re P E s s ( t, Σ) = { ε, 1 , 11 , 2 , 22 } , S E ss ( t, Σ) = { t, f ( x 1 , x 2 ) , x 1 , f ( f ( x 1 , x 2 ) , x 3 ) , x 3 } P F i c ( t, Σ ) = P os ( t ) \ P E ss ( t, Σ) = { 12 , 21 , 211 , 212 } , S F ic ( t, Σ) = { x 2 } . The Σ − essential a nd Σ − fictive pos itions of t are represented by large and small black circle s , resp ectively in Figur e 2. Note that | P F ic ( t, Σ) | > | S F ic ( t, Σ ) | . This is b ecause there is one subterm, f ( x 1 , x 2 ), whic h o ccurs mor e than o nce, o nce each in an essential and no n-essential p osition, so that | P os ( t ) | > | S ub ( t ) | . ✉ t ◗ ◗ ◗ ◗ ◗ ✑ ✑ ✑ ✑ ✑ ✉ ✉ ❅ ❅ ❅    ❆ ❆ ❆ ✁ ✁ ✁ ✉ r ✉ r ❆ ❆ ❆ ✁ ✁ ✁ r r 1 f 21 f x 3 211 x 2 x 2 x 1 11 12 x 1 212 f f 22 2 ε Figure 2. Σ − essential positio ns for t from Example 2.2. ESSENTIAL V ARIABLES AND POSITIONS IN T ERMS 7 Theorem 2.2. If p ∈ P E ss ( t, Σ) , then e ach p osition q ∈ P os ( t ) with q  p is Σ − essential for t . Pr o of. Let sub t ( q ) = s and su b t ( p ) = r . Now, q  p implies tha t r ∈ S ub ( s ) and S ub ( r ) ⊂ S ub ( s ) . Let n be a natura l n umber such that t ∈ W τ ( X n ) . F rom p ∈ P E ss ( t, Σ) it follows that there is a term v ∈ W τ ( X ) fo r which v 6 = x n +1 and Σ 6| = t ≈ t ( p ; x n +1 )( x n +1 ← v ) , i.e., Σ 6| = t ≈ t ( p ; v ) . Consequently , there is an algebra A = h A, F i o f t yp e τ such that A | = Σ and t A 6 = t ( p ; v ) A . Let m be a na tural num b er such that t ∈ W τ ( X m ) and v ∈ W τ ( X m ). Let ( a 1 , . . . , a m ) ∈ A m be a tuple suc h that t A ( a 1 , . . . , a n ) 6 = t ( p ; v ) A ( a 1 , . . . , a m ) . Let u ∈ W τ ( X m ) b e the ter m u = s ( q ′ ; v ) , where p = q ◦ q ′ and q ′ ∈ P os ( s ) . Hence w e have Σ | = t ( p ; v ) ≈ t ( q ; u ) and t A ( a 1 , . . . , a n ) 6 = t ( q ; u ) A ( a 1 , . . . , a m ) . Consequently t A 6 = t ( q ; u ) A , i.e., Σ 6| = t ≈ t ( q ; x n +1 )( x n +1 ← u ) and q ∈ P E ss ( t, Σ).  Corollary 2. 4. If q ∈ P F ic ( t, Σ) , then e ach p osition p ∈ P os ( t ) with q  p is Σ − fictive for t . Theorem 2.3. L et t ∈ W τ ( X ) b e a term of typ e τ and let Σ ⊆ I d ( τ ) b e a set of identities of typ e τ . If p ∈ P F ic ( t, Σ) , then Σ | = t ≈ t ( p ; v ) , for e ach t erm v ∈ W τ ( X ) . Pr o of. Let p ∈ P F i c ( t, Σ ) a nd let us supp ose tha t the theorem is fals e . Then there is a term v ∈ W τ ( X ) w ith v 6 = sub t ( p ), suc h that Σ 6| = t ≈ t ( p ; v ) . Let sub t ( p ) = r and let n be a na tural num b er, suc h that v , t ∈ W τ ( X n ). Then t ( p ; v ) = t ( p ; x n +1 )( x n +1 ← v ) and t = t ( p ; r ) = t ( p ; x n +1 )( x n +1 ← r ) . Our suppos ition shows that Σ 6| = t ≈ t ( p ; v ) ⇐ ⇒ Σ 6| = t ( p ; x n +1 )( x n +1 ← r ) ≈ t ( p ; x n +1 )( x n +1 ← v ) . Hence there is an algebra A = h A, F i and n + 2 elements a 1 , . . . , a n , a , b of A such that ( t ( p ; x n +1 )) A ( a 1 , . . . , a n , a ) 6 = ( t ( p ; x n +1 )) A ( a 1 , . . . , a n , b ) , where a = r A ( a 1 , . . . , a n ) and b = v A ( a 1 , . . . , a n ) . This means that x n +1 ∈ E ss ( t ( p ; x n +1 ) , Σ). Hence p ∈ P E ss ( t, Σ), whic h is a contradiction.  8 SL. SHTRAKO V Corollary 2.5. If p ∈ P os ( t ) is a Σ − fictive p osition for t , then p is fictive fo r t with re sp e ct to e ach algeb r a A with A | = Σ . Corollary 2.6. If p ∈ P E ss ( t, Σ) , t ∈ W τ ( X n ) , t hen p is essential for t with r esp e ct to e ach M od (Σ) − fr e e algebr a with at le ast n + 1 fr e e gener ators. If Σ | = t ≈ s a nd s ∈ S ub ( t ) is a prop er subter m of t , one might exp e ct that the po sitions of t which are “outside” of s hav e to b e Σ − fictive. T o see tha t this is not true, we co ns ider the s et of o per ations ∨ , ∧ and ¬ with type τ := (2 , 2 , 1 ). Let Σ b e the s et of ident ities satisfied in a Boolea n alg ebra. Then it is easy to prov e that if t = x 1 ∧ ( x 2 ∨ ¬ ( x 2 )), then we ha ve Σ | = t ≈ x 1 , but P E ss ( t, Σ) = P os ( t ) . Now, w e are going to generalize comp os ition of terms and t o describe the cor - resp onding deductiv e sy stems. Let Σ be a set of iden tities of type τ . Tw o terms t and s are called Σ -e quivalent (briefly , Σ -e qual ) if Σ | = t ≈ s . Definition 2.5. Let t , r , s ∈ W τ ( X ) and Σ S t r = { v ∈ S ub ( t ) | Σ | = r ≈ v } be th e set of all subterms of t which are Σ − equal to r . T erm Σ − composition of t and r b y s is defined as follows (i) t Σ ( r ← s ) = t if Σ S t r = ∅ ; (ii) t Σ ( r ← s ) = s if Σ | = t ≈ r , a nd (iii) t Σ ( r ← s ) = f ( t Σ 1 ( r ← s ) , . . . , t Σ n ( r ← s )) , if t = f ( t 1 , . . . , t n ). Let Σ P t r = { p ∈ P os ( t ) | su b t ( p ) ∈ Σ S t r } be the set of all p ositions of subterms of t which are Σ − eq ual to r . Let P t r = { p 1 , . . . , p m } b e the set of all the min imal elements in Σ P t r with r e spe ct to the ordering ≺ in the set of p ositions, i.e., p ∈ P t r if for each q ∈ P t r we hav e q 6≺ p . Let r j = su b t ( p j ) for j = 1 , . . . , m . Clear ly , t Σ ( r ← s ) = t ( P t r ; s ) . Example 2.3. Let us consider the set Σ of iden tities satisfied in the v a r iety RB of recta ng ular bands (see Exa mple 2.2). Let r = f ( x 1 , x 2 ) a nd let the terms t and u be the same as in Example 2.1. Then w e hav e Σ S t r = { f ( x 1 , x 2 ) , f ( f ( x 1 , x 2 ) , x 2 ) } , Σ P t r = { 12 11 , 121 } , P t r = { 12 1 } and t Σ ( r ← u ) = t (121; u ) (see Figure 1). So, the term t Σ ( r ← u ) is the t erm obtained from t b y replacing r by u at a ny minimal positions whose subterm is Σ-equal to r , where minimalit y refers to the order  on the set of p ositio ns. Prop ositio n 2.2. If Σ | = r ≈ v , and u, t ∈ W τ ( X ) then: (i) Σ | = t Σ ( u ← u ) ≈ t ; (ii) P t r = P t v ; (iii) Σ | = t Σ ( r ← u ) ≈ t Σ ( v ← u ) . Pr o of. (i) If t Σ ( u ← u ) = t , then the pr o p o sition is obvious. Let us assume that t Σ ( u ← u ) 6 = t . Hence Σ P t u \ P t u = { q 1 , . . . , q k } 6 = ∅ . Let P t u = { p 1 , . . . , p m } and ESSENTIAL V ARIABLES AND POSITIONS IN T ERMS 9 p i ∈ P t u . If p i 6≺ q j for all j ∈ { 1 , . . . , k } , then since Σ | = sub t ( p i ) ≈ u and D 5 we obtain Σ | = t ( p i ; u ) ≈ t. If p i ≺ q j , for some j ∈ { 1 , . . . , k } , then we hav e Σ | = sub t ( p i ) ≈ s u b t ( q j ) ≈ u . Since t = t ( p i ; s u b t ( p i )) = t ( p i ; s u b t ( p i ))( q j ; s u b t ( q j )) we obtain Σ | = t ( p i ; u ) ≈ t. Finally , w e hav e Σ | = t Σ ( u ← u ) ≈ t . (ii) and (iii) are clea r.  Corollary 2 . 7. (i) P t Σ ( u ← u ) u = Σ P t Σ ( u ← u ) u ; (ii) t Σ ( u ← u ) Σ ( u ← v ) = t Σ ( u ← u )( u ← v ) for any term v ∈ W τ ( X ) . Prop ositio n 2.3. If Σ | = t ≈ s and Σ | = r ≈ v , then P t r ⊆ P F ic ( t, Σ) ⇐ ⇒ P s v ⊆ P F ic ( s, Σ) . Next we consider a deductive system, whic h is based on the Σ - comp ositions of terms. 3. St abl e v arieties and globall y inv ariant congruences Our next g oal is to in tro duce deductiv e closures o n the subs e ts of I d ( τ ) whic h generate elements of the lattices L ( τ ) and E ( τ ). These closure s a re based on t wo concepts - satisfaction of an identit y by a v a riety a nd deduction of an identit y . Definition 3.1. [1] A s et Σ of ide ntities o f type τ is D − deductively closed if it sat- isfies the following axioms (some authors ca ll them “ deductive rules” , “ deriv atio n rules”, “pro ductions”, etc.): D 1 (reflexivity) t ≈ t ∈ Σ for each term t ∈ W τ ( X ); D 2 (symmetry) ( t ≈ s ∈ Σ) ⇒ s ≈ t ∈ Σ; D 3 (transitivity) ( t ≈ s ∈ Σ) & ( s ≈ r ∈ Σ) ⇒ t ≈ r ∈ Σ; D 4 (v ariable inductive substitution) ( t ≈ s ∈ Σ) & ( r ∈ W τ ( X )) ⇒ t ( x ← r ) ≈ s ( x ← r ) ∈ Σ; D 5 (term p ositional repla cement ) ( t ≈ s ∈ Σ) & ( r ∈ W τ ( X )) & ( su b r ( p ) = t ) ⇒ r ( p ; s ) ≈ r ∈ Σ. F or any set of identities Σ the smallest D − deductively closed set containing Σ is called the D − closur e of Σ a nd it is denoted by D (Σ) . Let Σ b e a s e t of ident ities of type τ . F or t ≈ s ∈ I d ( τ ) we say Σ ⊢ t ≈ s (“Σ D − proves t ≈ s ” ) if there is a sequence of iden tities ( D − deduction) t 1 ≈ s 1 , . . . , t n ≈ s n , suc h that each iden tity b elongs to Σ or is a r esult of applying any of the deriv a tion r ules D 1 − D 5 to previous iden tities in the sequence and the last ident ity t n ≈ s n is t ≈ s. According to [1], Σ | = t ≈ s if and only if t ≈ s ∈ D (Σ) and the closur e D (Σ) is a fully inv a riant congruence for each set Σ o f iden tities of a g iven t y pe. It is known that there exists a v ariety V ⊂ A l g ( τ ) with I d ( V ) = Σ if and only if Σ is a fully inv ar iant congruence (Theor em 14.17 [1]). Using pro per ties of the essential v a riables and positions w e can divide the r ules D 4 and D 5 int o four rules which distinguish b etw ee n oper ating with essen tial or fictive ob jects in the identities. 10 SL. SHTRAKO V Prop ositio n 3.1. A set Σ is D − de duct ively close d if it satisfies rules D 1 , D 2 , D 3 and D ′ 4 (essential varia ble inductive substitution) ( t ≈ s ∈ Σ) & ( r ∈ W τ ( X )) & ( x ∈ E ss ( t, Σ)) ⇒ t ( x ← r ) ≈ s ( x ← r ) ∈ Σ ; D ′′ 4 (fictive va riable inductive substitution) ( t ≈ s ∈ Σ) & ( r ∈ W τ ( X )) & ( x ∈ F ic ( t, Σ)) ⇒ t ( x ← r ) ≈ s ∈ Σ ; D ′ 5 (essential p ositional term r eplac ement) ( t ≈ s ∈ Σ) & ( su b r ( p ) = t, p ∈ P E ss ( r, Σ)) ⇒ r ( p ; s ) ≈ r ∈ Σ; D ′′ 5 (fictive p ositional term r eplac ement) ( t, s, r ∈ W τ ( X )) & ( sub r ( p ) = t, p ∈ P F ic ( r , Σ)) ⇒ r ( p ; s ) ≈ r ∈ Σ . W e will say that a set Σ o f iden tities is complete if D (Σ) = I d ( τ ). It is clear that if Σ is a complete set, then M od (Σ) is a trivial v ar ie ty . F or fictive po sitions in terms and complete se ts o f iden tities, w e hav e: Σ is complete ⇐ ⇒ ( ∀ t ∈ W τ ( X )) P os ( t ) = P F ic ( t, Σ) . Let Σ b e a non-co mplete set of identities. Then from Theor e m 2.1 a nd Theor em 2.3, it follo ws that w hen applying the rules D ′′ 4 and D ′′ 5 , we can skip these steps in the deduction pro cess, without any reflection o n the res ulting iden tities. Hence, if Σ is a no n-complete se t of iden tities, then Σ is D − deductively c lo sed if it satisfies the rules D 1 , D 2 , D 3 , D ′ 4 , D ′ 5 . In order to obtain new elements in the lattices L ( τ ) and E ( τ ), we have to extend the deriv a tio n rules D 1 − D 5 . Definition 3.2. A set Σ of identities is Σ R - deductively c lo sed if it satisfies the rules D 1 , D 2 , D 3 , D 4 and Σ R 1 ( Σ r eplac ement) ( t ≈ s, r ≈ v , u ≈ w ∈ Σ)& ( r ∈ S E ss ( t, Σ)) & ( v ∈ S E ss ( s, Σ)) ⇒ t Σ ( r ← u ) ≈ s Σ ( v ← w ) ∈ Σ . F or any set of identities Σ the smallest Σ R − deductively clo sed set con taining Σ is called Σ R − c losure of Σ and it is denoted by Σ R (Σ) . Let Σ b e a set of iden tities of t y pe τ . F or t ≈ s ∈ I d ( τ ) we say Σ ⊢ Σ R t ≈ s (“Σ Σ R -proves t ≈ s ” ) if there is a s equence o f identit ies t 1 ≈ s 1 , . . . , t n ≈ s n , such that each identit y belo ngs to Σ or is a result of a pplying any of the der iv ation r ules D 1 , D 2 , D 3 , D 4 or Σ R 1 to previous iden tities in the sequence and the las t identit y t n ≈ s n is t ≈ s. Let t ≈ s be an identit y a nd A b e an algebra of t y p e τ . A | = Σ R t ≈ s means that A | = t Σ ( r ← v ) ≈ s Σ ( r ← v ) for e very r ∈ S E ss ( t, Σ) ∩ S E ss ( s, Σ) and v ∈ W τ ( X ). Let Σ b e a set of identities. F or t, s ∈ W τ ( X ) we say Σ | = Σ R t ≈ s (read: “Σ Σ R − yields t ≈ s ”) if, given any alg ebra A , A | = Σ R Σ ⇒ A | = Σ R t ≈ s. ESSENTIAL V ARIABLES AND POSITIONS IN T ERMS 11 Theorem 3.1. Σ R i s a closur e op er ator in the set I d ( τ ) , i.e., (i) Σ ⊆ Σ R (Σ) ; (ii) Σ 1 ⊆ Σ 2 ⇒ Σ 1 R (Σ 1 ) ⊆ Σ 2 R (Σ 2 ) ; (iii) Σ R (Σ R (Σ)) = Σ R (Σ) . The following lemma is clea r. Lemma 3 . 1. F or e ach set Σ ⊆ I d ( τ ) and fo r e ach identity t ≈ s ∈ I d ( τ ) we have Σ ⊢ Σ R t ≈ s ⇐ ⇒ Σ R (Σ) ⊢ t ≈ s. Theorem 3.2. (The Completeness The or em for Σ R - Equational L o gic) L et Σ ⊆ I d ( τ ) b e a set of identities and t ≈ s ∈ I d ( τ ) . Then Σ | = Σ R t ≈ s ⇐ ⇒ Σ ⊢ Σ R t ≈ s. Pr o of. The implication Σ ⊢ Σ R t ≈ s ⇒ Σ | = Σ R t ≈ s follows by Σ ⊢ Σ R t ≈ s ⇒ t ≈ s ∈ Σ R (Σ) since we ha ve use d only pro per ties under which Σ R (Σ) is closed, i.e., under D 1 , D 2 , D 3 , D 4 and Σ R 1 . F or the conv e r se of this, let us note that for t ∈ W τ ( X ) we have Σ ⊢ Σ R t ≈ t and if t ≈ s ∈ Σ then Σ ⊢ Σ R t ≈ s . If Σ ⊢ Σ R t ≈ s , then there is a formal Σ R − deduction t 1 ≈ s 1 , . . . , t n ≈ s n of t ≈ s. But then t 1 ≈ s 1 , . . . , t n ≈ s n , s n ≈ t n is a Σ R − deduction of s ≈ t. If Σ ⊢ Σ R t ≈ s and Σ ⊢ Σ R s ≈ r let t 1 ≈ s 1 , . . . , t n ≈ s n be a Σ R − de ductio n o f t ≈ s and let s 1 ≈ r 1 , . . . , s k ≈ r k be a Σ R − deduction of s ≈ r. Then t 1 ≈ s 1 , . . . , t n ≈ s n , s 1 ≈ r 1 , . . . , s k ≈ r k , t n ≈ r k is a Σ R − deduction of t ≈ r . Hence Σ ⊢ Σ R t ≈ r. Let Σ | = Σ R t ≈ s , Σ | = Σ R r ≈ v , Σ | = Σ R u ≈ w and Σ | = Σ R t Σ ( r ← u ) ≈ s Σ ( v ← w ). Suppo s e that Σ ⊢ Σ R t ≈ s , Σ ⊢ Σ R r ≈ v and Σ ⊢ Σ R u ≈ w . Let t 1 ≈ s 1 , . . . , t n ≈ s n , r 1 ≈ v 1 , . . . , r m ≈ v m and u 1 ≈ w 1 , . . . , u k ≈ w k be Σ R − deductions of t ≈ s , r ≈ v and u ≈ w . Then t 1 ≈ s 1 , . . . , t n ≈ s n , r 1 ≈ v 1 , . . . , r m ≈ v m , u 1 ≈ w 1 , . . . , u k ≈ w k , t Σ n ( r m ← u k ) ≈ s Σ n ( v m ← w k ) is a Σ R − deduction of t Σ ( r ← u ) ≈ s Σ ( v ← w ). Hence Σ ⊢ Σ R t Σ ( r ← u ) ≈ s Σ ( v ← w ).  Theorem 3.3 . F or e ach set of identities Σ the closur e Σ R (Σ) is a ful ly invariant c ongruenc e, but Σ R (Σ) is not in gener al e qual to D (Σ) . Pr o of. Let Σ be a Σ R − deductively closed set of iden tities. W e will prov e tha t Σ is a fully in v ar iant cong ruence. It ha s to be s hown that Σ satis fie s the rule D 5 , i.e., if r ∈ W τ ( X ), t ≈ s ∈ Σ and p ∈ P os ( r ), then r ( p ; t ) ≈ r ( p ; s ) ∈ Σ . If p / ∈ P E ss ( r, Σ), then according to Prop osition 3.1 we have Σ | = r ( p ; v ) ≈ r ( p ; w ) for a ll terms v , w ∈ W τ ( X ). 12 SL. SHTRAKO V Let p ∈ P E ss ( r, Σ) and let n b e a natural n umber suc h that r, t, s ∈ W τ ( X n ) and let us consider the term u = r ( p ; x n +1 ). Clearly , u ≈ u ∈ Σ, b ecaus e of D 1 . W e have u Σ ( x n +1 ← v ) = u ( x n +1 ← v ) = u ( p ; v ) for each v ∈ W τ ( X ) . No w fro m Σ R 1 we obtain u Σ ( x n +1 ← t ) ≈ u Σ ( x n +1 ← s ) ∈ Σ , i.e., u ( x n +1 ← t ) ≈ u ( x n +1 ← s ) ∈ Σ , and r ( p ; t ) ≈ r ( p ; s ) ∈ Σ. F urthermor e, we will pro duce a fully inv ariant congruence Σ , whic h is not Σ R − deductively closed. Let us consider the v ariety S G = M od (Σ) of semigr oups, where Σ = { f ( x 1 , f ( x 2 , x 3 )) ≈ f ( f ( x 1 , x 2 ) , x 3 ) } . F rom Theor em 14 .17 of [1] it follows tha t if Σ is a fully inv aria nt congruence, then D (Σ) = I d ( M od (Σ)) . Hence I d ( S G ) = D (Σ) . Let t = f ( f ( f ( x 1 , x 2 ) , x 1 ) , x 2 ) and s = f ( f ( x 1 , x 2 ) , f ( x 1 , x 2 )) . It is not difficult to see that Σ | = t ≈ s , i.e., t ≈ s ∈ D (Σ). Let us set r = v = f ( x 1 , x 2 ) and u = w = x 1 . Clearly , for e ach z ∈ W τ ( X ) w e have P E ss ( z , Σ) = P os ( z ) a nd r ∈ S E ss ( t, Σ), and v ∈ S E ss ( s, Σ). Since P t r = { 11 } and P v s = { 1 , 2 } , we obtain t Σ ( r ← u ) = f ( f ( x 1 , , x 1 ) , x 2 ) and s Σ ( v ← w ) = f ( x 1 , x 1 ) . Hence Σ 6| = t Σ ( r ← u ) ≈ s Σ ( v ← w ) . On the other side w e ha ve t Σ ( r ← u ) ≈ s Σ ( v ← w ) ∈ Σ R (Σ). Consequently , D (Σ) is a prop er subset (equational theory) of Σ R (Σ) and M od (Σ R (Σ)) is a prop er sub v ariety of S G , which co nt ains the v ariety RB of r ectangular bands as a subv ar iety , acco rding to Example 3.1, b elow.  Lemma 3 . 2. F or e ach set Σ ⊆ I d ( τ ) and fo r e ach identity t ≈ s ∈ I d ( τ ) we have Σ | = Σ R t ≈ s ⇐ ⇒ Σ R (Σ) | = t ≈ s. The lemma follows immediately from Lemma 3.1 and Theor em 3 .2. Definition 3.3. A set of identities Σ is called a glo bally in v ariant congr uence if it is Σ R − deductively closed. A v arie ty V of t yp e τ is called stable if I d ( V ) is Σ R − deductively c lo sed, i.e., I d ( V ) is a glo bally in v ar iant congruence. Note that when Σ is a globally inv ariant congruence it is poss ible to apply substitutions or replacements in any place ( op era tion sym b ol) of terms whic h explains the word “ globally”. Example 3.1. Now, we will produce a fully inv ariant congr uence Σ, which is a glo bally inv ariant congruence . Let us consider the v a riety RB = M od (Σ) o f rectangular bands, where Σ is defined a s in Example 2.2. ESSENTIAL V ARIABLES AND POSITIONS IN T ERMS 13 The set Σ consists of all equatio ns s ≈ t such that the first v ar iable (leftmost) of s a grees with the first v ariable o f t , i.e., l e f tmost ( t ) = l ef tmost ( s ) and the la st v aria ble (rig h tmost) o f s a g rees with the las t v ar iable of t , i.e., rig htmost ( t ) = rig htmost ( s ). It is w ell known that Σ is a fully inv a riant congruence and a totally inv ar iant congruence ( see [2, 4]). F rom Theorem 14.17 of [1] it follo ws that I d ( R B ) = D (Σ) . Let t, s, r, v , u, w ∈ W τ ( X ) be six terms such that Σ | = t ≈ s, Σ | = r ≈ v , Σ | = u ≈ w , r ∈ S E ss ( t, Σ) and v ∈ S E ss ( s, Σ). Thu s we hav e l ef tmost ( t ) = l ef tmost ( s ) , l ef tmost ( r ) = l ef tmost ( v ) , l ef tmost ( u ) = l ef tmost ( w ) , rig htmost ( t ) = r ig htm ost ( s ) , rig htmost ( r ) = r ig htm ost ( v ) , r ig htmost ( u ) = r ig h tmost ( w ) . F rom r ∈ S E ss ( t, Σ) a nd v ∈ S E ss ( s, Σ) ( see Example 2.2), we obtain l ef tmost ( t ) = l ef tmost ( r ) , l ef tmost ( s ) = l ef tmost ( v ) or rig htmost ( t ) = r ig htm ost ( r ) , r ig htmost ( s ) = r ig htmost ( v ) . Hence l ef tmost ( t Σ ( r ← u )) = l ef tmost ( s Σ ( v ← w )) and rig htmost ( t Σ ( r ← u )) = r ig htmost ( s Σ ( v ← w )) . W e a re going to compar e globally inv ar iant congruences with the to tally inv ari- ant cong r uences, defined b y hypersubstitutions. In [2, 4] the solid v arieties are defined by adding a new der iv ation rule which uses the concept of hypers ubstitutions. Let σ : F → W τ ( X ) be a mapping which assig ns to every op eration symbol f ∈ F n an n − ary term. Suc h ma ppings are called hyp ersubstit u tions (of t yp e τ ). If one replaces every oper ation symbol f in a giv en term t ∈ W τ ( X ) b y the term σ ( f ), then the resulting term ˆ σ [ t ] is the image of t under the extens io n ˆ σ on the set W τ ( X ). The mono id of all hypersubstitutions is denoted by H y p ( τ ) . Let Σ b e a s et o f identities. The hypersubstitution deriv ation rule is defined as follows: H 1 (hyp ersubstitut ion) ( t ≈ s ∈ Σ & σ ∈ H yp ( τ )) ⇒ ˆ σ [ t ] ≈ ˆ σ [ s ] ∈ Σ . A set Σ is called χ − de ductively close d (hyp er e quational t he ory, or total ly in- variant c ongruenc e) if it is clo sed with resp ect to the rules D 1 , D 2 , D 3 , D 4 , D 5 and H 1 . The χ − closur e χ (Σ) of a set Σ of identities is defined in a natural w ay and the meaning of Σ | = χ and Σ ⊢ χ is clear. It is ob vious that D (Σ) ⊆ χ (Σ) for each s et of iden tities Σ ⊆ I d ( τ ). There are examples of Σ such that D (Σ) 6 = χ (Σ), whic h shows that the co rresp onding v ariety M od ( χ (Σ)) is a pr op er sub v ar iet y of M od ( D (Σ)). A v ariety V for whic h I d ( V ) is χ − deductively closed is called solid v ariety of type τ [2]. 14 SL. SHTRAKO V A more co mplex closure op erator on sets of iden tities is studied in [3]. This op erator is based on the concept o f coloured terms and mu lti-hyper s ubstitutions. The next prop osition deals w ith the relations betw een the clos ure op era to rs Σ R and χ . Prop ositio n 3.2. T her e exists a stable variety, whi ch is n ot a solid va riety. Pr o of. Let us co nsider t he type τ = (2) and Σ = { f ( f ( x 1 , x 2 ) , x 1 ) ≈ f ( x 1 , x 1 ) } . W e will show that LA = M od (Σ) is a stable v ariety . So, we hav e to prove that (1) Σ | = t Σ ( r ← u ) ≈ s Σ ( v ← w ) , when Σ | = t ≈ s , Σ | = r ≈ v , Σ | = u ≈ w , r ∈ S E ss ( t, Σ) and v ∈ S E ss ( s, Σ). W e will pro ceed by induction on De pth ( t ) - the depth of the term t . The case D epth ( t ) = 0 is trivial. Let D epth ( t ) = 1. If t = f ( x 1 , x 2 ) then Σ | = t ≈ s implies s = f ( x 1 , x 2 ) and (1) is satisfied in this ca se. Let us co nsider the case t = f ( x 1 , x 1 ). If s = f ( x 1 , x 1 ) then clearly (1) holds. Let s = f ( f ( x 1 , s 1 ) , x 1 ) for some s 1 ∈ W τ ( X ). Since the p ositions in s 1 are Σ-fictive in s it follows that v can b e one of the terms x 1 or s . O n the o ther side we hav e S E s s ( t, Σ) = { x 1 , t } . Hence (1) is satisfied, again. Our inductiv e supp osition is that if D epth ( t ) < k then (1) is s a tisfied for all s, r , u, v , w ∈ W τ ( X ) with Σ | = t ≈ s , Σ | = r ≈ v , Σ | = u ≈ w , r ∈ S E ss ( t, Σ) and v ∈ S E ss ( s, Σ). Let Dep th ( t ) = k > 1 and D epth ( s ) ≥ k . Then we hav e t = f ( t 1 , t 2 ) and s = f ( s 1 , s 2 ), such that t 1 or t 2 is not a v ar iable. If Σ | = t ≈ r or Σ | = s ≈ v , then b y Definition 2.5 ( ii ) and the transitivit y D 3 , it follows that (1) is Σ | = u ≈ w and we a re done. Next, we as sume that Σ 6| = t ≈ r and Σ 6| = s ≈ v . First, let t 1 ∈ X . Then s 1 = t 1 and Σ | = t 2 ≈ s 2 . Thus, from the inductive suppo sition it follows that (1) is satisfied. Second, let t 1 / ∈ X . Then we hav e s 1 / ∈ X , also. Hence t = f ( f ( t 11 , t 12 ) , t 2 ), s = f ( f ( s 11 , s 12 ) , s 2 ) and Σ | = t 2 ≈ s 2 . Let Σ | = t 11 ≈ t 2 and Σ | = s 11 ≈ s 2 . Then w e hav e Σ | = t ≈ f ( t 2 , t 2 ) a nd Σ | = f ( t 2 , t 2 ) ≈ f ( s 2 , s 2 ). On the other side all positions in t 12 and s 12 are Σ-fictive in t and s , res pectively . Th us we ha ve Σ | = t Σ ( r ← u ) ≈ f ( t Σ 2 ( r ← u ) , t Σ 2 ( r ← u )) and Σ | = s Σ ( v ← w ) ≈ f ( s Σ 2 ( v ← w ) , s Σ 2 ( v ← w )) . Hence (1) is satisfied, in this case, again. If Σ | = t 11 ≈ t 2 and Σ 6| = s 11 ≈ s 2 then we hav e Σ | = f ( t 2 , t 2 ) ≈ f ( s 1 , s 2 ) and Σ | = s 1 ≈ s 2 ≈ t 2 . This implies that (1) is satisfied, again. Let Σ 6| = t 11 ≈ t 2 and Σ | = t 1 ≈ t 2 . Then w e ha ve Σ | = t ≈ f ( t 1 , t 2 ). Now, we pro ceed similarly as in the case Σ | = t 11 ≈ t 2 and Σ 6| = s 11 ≈ s 2 . If Σ 6| = t 11 ≈ t 2 ESSENTIAL V ARIABLES AND POSITIONS IN T ERMS 15 and Σ 6| = t 1 ≈ t 2 , then we have Σ 6| = s 1 ≈ s 2 . Hence Σ | = t 1 ≈ s 1 and Σ | = t 2 ≈ s 2 . Again, from the inductive suppositio n w e prov e (1). T o pr ov e tha t LA is not a solid v ar ie t y , let u s co nsider the following terms y = f ( f ( x 1 , x 2 ) , x 1 ) and z = f ( x 1 , x 1 ). Let σ ∈ H yp ( τ ) b e the h yp ers ubstitution, defined as follows: σ ( f ( x 1 , x 2 )) := f ( x 2 , x 1 ). It is clea r tha t Σ | = y ≈ z . On th e other side we have ˆ σ [ y ] = f ( x 1 , f ( x 2 , x 1 )) and ˆ σ [ z ] = f ( x 1 , x 1 ). Th us, w e obtain Σ 6| = ˆ σ [ y ] ≈ ˆ σ [ z ]. Hence LA is not a solid v a riety .  Remark 3.1. By analo gy , it follows that the v ariety RA = M od ( { f ( x 1 , f ( x 2 , x 1 )) ≈ f ( x 1 , x 1 ) } ) is stable, but not solid, also . The v arieties of left-zero bands L 0 = M od ( { f ( x 1 , x 2 ) ≈ x 1 } ) a nd of right-zero bands R 0 = M od ( { f ( x 1 , x 2 ) ≈ x 2 } ) are other examples of s table v arieties, whic h are not solid ones. W e do not know whether there is a non-tr ivial solid v ariety which is not stable? 4. Σ − balanced identities and simplifica tion of deductions Regular identit ies [1, 4] are identities in which the sa me v aria ble s o ccur on each side of the ident ity . B alanced identities are identit ies in which each v ariable o ccurs the same num b er of times o n each side o f th e identit y . In an analog ous w ay we c o nsider the concept of Σ − ba lanced iden tities. Let t, r ∈ W τ ( X ) be t wo terms of type τ and Σ ⊂ I d ( τ ) be a set of identities. E P t r denotes the set of all Σ-esse n tial p ositions from P t r , i.e., E P t r = P E ss ( t, Σ) ∩ P t r . Definition 4.1 . Let Σ ⊂ I d ( τ ). W e will s ay that an identit y t ≈ s of t yp e τ is Σ − b alanc e d if | E P t q | = | E P s q | for all q ∈ W τ ( X ). Example 4.1. Let Σ b e the set of identities satisfied in the v ariety RB of r ect- angular bands (see Example 2.2). Let us consider the following three terms t = f ( f ( x 1 , x 2 ) , f ( x 1 , x 3 )), s = f ( x 1 , f ( f ( x 1 , x 2 ) , x 3 )) and r = f ( f ( f ( x 1 , f ( x 3 , x 2 )) , x 3 ) , f ( x 1 , x 3 )). Clear ly , Σ | = t ≈ s , Σ | = t ≈ r , S E ss ( t, Σ) = S E ss ( r, Σ) = { x 1 , f ( x 1 , x 2 ) , f ( x 1 , x 3 ) , x 3 } and S E ss ( s, Σ) = { x 1 , f ( x 1 , x 3 ) , x 3 } . Th us we have E P t x 1 = { 11 } , E P t x 3 = { 22 } , E P t f ( x 1 ,x 2 ) = { 1 } , E P t f ( x 1 ,x 3 ) = { ε } , E P t t = { ε } , E P s x 1 = { 1 } , E P s x 3 = { 22 } , E P s f ( x 1 ,x 3 ) = { ε } , E P s s = { ε } , and E P r x 1 = { 111 } , E P r x 3 = { 22 } , E P r f ( x 1 ,x 2 ) = { 11 } , E P r f ( x 1 ,x 3 ) = { ε } , E P r r = { ε } . Hence the ident ity t ≈ r is Σ − balanced, but t ≈ s is not Σ − bala nced. Theorem 4.1. L et Σ ⊂ I d ( τ ) b e a set of Σ − b alanc e d identities. If t her e is a Σ R − de duction of t ≈ s with Σ − b alanc e d identities, then t ≈ s is a Σ − b alanc e d identity of typ e τ . 16 SL. SHTRAKO V Pr o of. Let t, s, r ∈ W τ ( X ) and let t ≈ s and s ≈ r be tw o Σ − bala nced iden tities of t yp e τ . Then for each term q ∈ W τ ( X ) w e have | E P t q | = | E P s q | and | E P s q | = | E P r q | . Hence | E P t q | = | E P r q | which shows that the identit y t ≈ r is Σ − balanced, to o . Let t ≈ s is a Σ − bala nc e d identit y in Σ R (Σ) and let r ∈ W τ ( X ) b e a term with t ∈ S E ss ( r, Σ) and sub r ( p ) = t . W e have Σ | = Σ R r ( p ; s ) ≈ r . F ro m Prop osition 2.2, w e obta in ( E P t q = E P s q & t ∈ S E ss ( r, Σ)) ⇒ E P r q = E P r ( p ; s ) q for all q ∈ W τ ( X ). Conseq uen tly the identit y r ( p ; s ) ≈ r is Σ − balanced, to o. Let t ≈ s , r ≈ v a nd u ≈ w b e Σ − balanced ident ities from Σ R (Σ). W e hav e to prov e that the resulting iden tit y t Σ ( r ← u ) ≈ s Σ ( v ← w ) is Σ − balanced. This will be done by induction on the depth (also called “heig h t” by some authors). (i) T he basis of induction is D ep th ( t ) = 1 (the case D epth ( t ) = 0 is trivial). Let t = f ( x 1 , . . . , x n ) ∈ W τ ( X n ) and le t s = g ( s 1 , . . . , s m ) . Hence, if r = x i for some i ∈ { 1 , . . . , n } , then E P t r = { i } a nd | E P t r | = | E P s v | = 1. (ia) If Σ | = r ≈ t , then Σ | = v ≈ s and we ha ve t Σ ( r ← u ) = u a nd s Σ ( v ← w ) = w . Hence t Σ ( r ← u ) ≈ s Σ ( v ← w ) is Σ − balanced in this case. (ib) Let Σ 6| = r ≈ x i for e a ch x i ∈ X n , i.e., r 6∈ X n and Σ 6| = r ≈ t . If r / ∈ X n ∪ { t } , then E P t r = E P s v = ∅ . Thus we have t Σ ( r ← u ) = t and s Σ ( v ← w ) = s and the resulting identit y t ≈ s is Σ − ba lanced. (ic) Let Σ | = r ≈ x i for some x i ∈ X n and Σ 6| = r ≈ t . W e hav e t Σ ( r ← u ) = t Σ ( x i ← u ) and s Σ ( v ← w ) = s Σ ( x i ← w ) . Then E P t Σ ( x i ← u ) q = E P u q and E P s Σ ( x i ← w ) q = E P w q for eac h q ∈ W τ ( X ), i.e., the resulting identit y t Σ ( r ← u ) ≈ s Σ ( v ← w ) is Σ − balanced, again. (ii) L et D epth ( t ) > 1, t = f ( t 1 , . . . , t n ) and s ∈ W τ ( X ), such that the iden tit y Σ | = t ≈ s is Σ-balanced. Supp ose that for each t ′ ∈ S E ss ( t, Σ) with t ′ 6 = t the following is true : if Σ | = t ′ ≈ s ′ is Σ-bala nce d iden tity for some s ′ ∈ W τ ( X ), then t ′ Σ ( r ← u ) ≈ s ′ Σ ( v ← w ) is Σ-balanced, also. (iia) Let Σ | = r ≈ t . Then we hav e E P t Σ ( x i ← u ) q = E P u q and E P s Σ ( x i ← w ) q = E P w q for each q ∈ W τ ( X ) and t he r esulting iden tity is Σ − balanced in that case, aga in. (iib) Let Σ 6| = r ≈ t and Σ | = r ≈ t i for some i = 1 , . . . , n. As in the case ( ic ) it can be pro ved that the identit y t Σ ( r ← u ) ≈ s Σ ( v ← w ) is Σ − balanced. (iic) Let Σ 6| = r ≈ t , Σ 6| = r ≈ t i for each i = 1 , . . . , n and there is j ∈ { 1 , . . . , n } with E P t j r = { r j 1 , . . . , r j k j } 6 = ∅ . Without loss o f generalit y a ssume that all suc h j are the natura l n umbers from the set L = { 1 , . . . , l } with l ≤ n. ESSENTIAL V ARIABLES AND POSITIONS IN T ERMS 17 Let j ∈ L . If Σ | = t ≈ t j is a Σ − balanced identit y , then Σ | = s ≈ t j is Σ − balanced, also a nd b y our assumption, w e ha ve that t Σ j ( r ← u ) ≈ s Σ ( v ← w ) is Σ − balanced iden tity . Hence w e hav e | E P t q | = | E P t j q | for all q ∈ W τ ( X ) . This implies that the resulting identit y is Σ − bala nced in this case, also. If Σ 6| = t ≈ t j for all j ∈ L then since t ≈ s is a Σ − balance d identit y , there are subter ms s 1 , . . . , s l of s such that Σ | = t j ≈ s j . According to our inductive suppo sition the last identities are Σ − balanced. Consequently , | E P t j r | = | E P s j r | , E P t r = ∪ l j =1 E P t j r and E P s v = ∪ l j =1 E P s j v . Hence t Σ ( r ← u ) ≈ s Σ ( v ← w ) is a Σ − balanced iden tit y .  The co mplexit y o f the pro blem of deduct ion de p ends on the complexit y of the algorithm fo r chec king when a p osition of a ter m is essential or not with re s pec t to a set of identities. The complexity of that algo rithm for finite alg ebras is discussed in [5], but it is bas ed on the full exhaustion of all p ossible cas es. There should b e a case or ca ses, when the pro ce s s of deduction can b e effectively simplified. This is, for instance, w he n a v ar ia ble x do es not b elong to v a r ( t ) and Σ | = t ≈ s. Therefore w e obtain x / ∈ E ss ( t, Σ) ∪ E ss ( s, Σ)(see Theorem 2.1). Then we ca n skip the rules D ′′ 4 and D ′′ 5 , ac c o rding to Propo sition 3.1. Obviously , it is very easy to chec k if x ∈ v a r ( t ) or not. References [1] S. Burri s, H. Sank appana v ar, A Course in Universal Algebr a , The mill ennium edition, 2000 [2] K.D enec ke, D.Lau, R.P¨ osc hel, D.Sch weigert, Solidifyable Clones, General Algebra 20, Hel- dermann V erl ag , Berlin 1993, pp.41-69. [3] K. Deneck e, J. Koppitz, Sl . Sh trak ov, M ulti-Hyp e rsubstitutions and Colour e d Solid V ariet ies , J. Al gebra and Computation, V olume 16, Number 4, August, 2006, pp.797-815. [4] E. Gracz´ ynsk a, On Normal and R e gular Identities and Hyp erident i ties , Universal and Ap- plied Al gebra, T uraw a, Poland 3 - 7 May 1988, W orld Scien tific (1989), 107-135. [5] Sl. Shtrak ov, K. Denec ke, Essential V ariables and Sep ar able Sets in Universal A lgebr a , J. Multi. V al. Logic, 2002, vol. 8(2), pp 165-181. [6] W. T aylor, Hy p eridentit ies and Hyp ervarieties, Aequatione s Mathematicae, 23(1981) , 30-49. E-mail addr ess : shtrakov@aix .swu.bg Dep ar tment of Computer Science, South-West University, 270 0 Blagoevgrad, Bul- garia URL : http://h ome.swu.bg/s htrakov