An Algebraic Characterisation of First-Order Logic with Neighbour

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( ,b c b, a, ) A, X and ) nfc,ti aalls between parallelism this fact, In . , L , ( enspeieyalrglrlan- regular all precisely defines min N bet x r lofrua whenever formulas also are .Both ). = ) FO( ) } hr r agae expressible languages are There . ¬ ↔ ) MSO( c , N ∗ max sepesbei h logic the in expressible is lodfie l eua lan- regular all defines also ,< A, abc h constants the , X MSO( ∗ , N ∃ ( bet A, ) L bet y Xϕ ))) xed oterquanti- their to extends +1) and and ) is ∧ A, enspeieyall precisely defines stu fteei a is there if true is X yBüchi-Elgot- by , not x N bet min ( ϕ min = 1 + ic both Since . o example, For . min r definable are , enbein definable max ) ∧¬ and , X y ϕ N and ) MSO ( max both min min max ) sa is ake bet is , - ) FO(A, +1) and FO(A, min, max, N ). For t > 0, we define isation of FO(A, +1) is a deep result [1], [5], [15], [26] in the equality with threshold t on the set N of natural numbers the theory of finite semigroups. The first observation is that by LTT is a variety of languages — a class of languages that

t i = j if i < t, is closed under Boolean operations, quotients with respect to i = j := (1) words, and inverse images under homomorphisms. Eilenberg’s (i ≥ t and j ≥ t otherwise. variety theorem states that varieties of languages correspond + + The word y ∈ A is a factor of the word u ∈ A if u = xyz to pseudovarieties of semigroups — a pseudovariety of fi- ∗ for some x, z in A . We use ♯(u,y) to denote the number of nite semigroups is a set of finite semigroups that is closed times the factor y appears in u, i.e. the number of pairs (x, z), under finite direct products, subsemigroups and quotients. ∗ where x, z ∈ A , such that u = xyz. The characterisation of LTT proceeds by first writing the ◮ Definition 1. Let ≈t , for k,t > 0, be the equivalence on class as a semidirect product of pseudovarieties of aperiodic k Acom A∗, whereby two words u and v are equivalent if either they commutative monoids ( ) and righty trivial semigroups D both have length at most k − 1 and u = v, or otherwise they ( is the pseudovariety of semigroups that satisfy the equation have se = e for all elements s and idempotents e). This step is the 1) the same prefix of length k − 1, algebraic analogue of synthesising an automaton for an LTT 2) the same suffix of length k − 1, language as a cascade of a scanner and an acceptor [1]. The 3) and the same number of occurrences, up to threshold t, next step in the proof uses the framework of categories for for all factors of length ≤ k, i.e. ♯(u,y) =t ♯(v,y) for obtaining the identities capturing the semidirect product [21], each word y ∈ A+ of length at most k. [22], [25]. In the case of wLRTT, this approach does not work; A language is locally threshold testable (or LTT for short) if because wLRTT does not form a variety of languages — t ∗ ∗ it is a unionof ≈k classes, for some k,t > 0. Locally threshold for instance, the language (xy) ab(xy) over the alphabet testable languages are precisely the class of languages defin- {a,b,x,y} is in wLRTT. Its inverse image under the mor- able in FO(A, +1) [1], [23]. phism a 7→ a,b 7→ b,c 7→ xy is the language c∗abc∗, Since the neighbour predicate N is definable using the over the alphabet {a,b,c}. As we have seen, this language successor relation in first-order logic, FO(A, min, max, N ) is not in wLRTT. Therefore, wLRTT is not closed under definable languages are a subset of LTT. But this inclusion is inverse image of morphisms, and hence not a variety of strict, as we have seen. languages. This necessitates a rubric, of the like of varieties r t We define a coarser equivalence ≈k by the counting factors and operations on varieties, to study the class wLRTT. only up to reverse. Let ♯r(w, v) denote the number of occurrences of v or vr in w, i.e. the number of pairs (x, y), where Our results. It was already observed in [8] that one needs x, y ∈ A∗, such that w = xvy or w = xvry. to extend semigroups with an involution (also called ⋆- ◮ Definition 2. Let k,t > 0. Two words w, w′ ∈ A∗ are semigroups) — an involution on a semigroup S is an operation ⋆ ⋆ ⋆ ⋆ ⋆ r t ′ ′ ⋆ such that (a ) = a and (ab) = b a , for each a,b ∈ S ≈k-equivalent if |w| < k and w = w , or w, w both are of length at least k, and they have — to characterise wLRTT. The involution operation is a gen- eralisation of the reversal operation on words. An involutory 1) the same prefix of length k − 1, alphabet A is a finite alphabet A with a bijection † on it. The 2) the same suffix of length k − 1, and map † extends to words over A as (a ...a )† = a† ...a†, 3) ♯r(w, v) =t ♯r(w′, v) for each word v ∈ A+ of length 1 n n 1 where each a ∈ A. This map is an involution on the semi- at most k. i group A+. We lift the notions of recognisability and syntactic It is shown in [8] that a language is definable in FO(A, N ) semigroups to the case of languages over involutory alphabets. if it is closed under the reverse operation and is a union of Syntactic algebras for such languages are semigroups with an r t equivalence classes of ≈k for some k,t > 0. Such languages involution. A finite alphabet A can be seen as an involutory are called locally-reversible threshold testable languages. The alphabet with the identity function id as †; such alphabets class of languages accepted by FO(A, min, max, N ) are are called hermitian. This just means that A+ is equipped characterised in a similar manner. A language L is weakly with the reverse operation as involution. Syntactic algebras locally-reversible threshold testable, wLRTT for short, if it is for languages over hermitian alphabets are semigroups with r t a union of equivalence classes of ≈k for some k,t > 0. By involutions that are generated by their hermitian elements (a a straight-forward adaptation of the proofs in [8], [9] we can is hermitian if a⋆ = a). show that Assume (A, †) and (B,⋆) are involutory alphabets and h: A+ → B+ is a morphism. The morphism h is involutory if ◮ Theorem 1. A language is definable in the logic h(w†)= h(w)⋆. An involution variety of languages maps each FO(A, min, max, N ) if and only if it is weakly locally- involutory alphabet A = (A, †) to a class of languages over A reversible threshold testable. that is closed under involutions, Boolean operations, quotients Obtaining an algebraic characterisation for the class with respect to words and inverse images under involutory wLRTT is an interesting problem. The analogous character- morphisms. We establish an Eilenberg-type correspondence between involution varieties of languages and pseudovarieties study of pseudovarieties of finite involution semigroups or of finite involution semigroups — classes of finite involution results similar to ours. semigroups that are closed under finite direct products, sub-⋆- semigroups and quotients. Orgranisation of the paper. In Section II, we describe involution semigroups and extend the notions of recognisability If S = (S,⋆) and T = (T, †) are involution semigroups, a two-sided action of T on S is compatible with the involution by semigroups to the case of languages with an involution. In Section III, involutory semidirect products are defined, and we if (tst′)⋆ = t′†s⋆t†, where tst′ denotes the left action by introduce the notion of locally hermitian semidirect product. t ∈ T and right action by t′ ∈ T on the element s ∈ S. A compatible two-sided action of T on S is locally hermitian Then, it shown that a language is wLRTT if and only if it is recognised by the locally hermitian semidirect product if for all idempotents e ∈ T and elements s ∈ S it holds of an aperiodic commutative semigroup and a locally trivial that ese† = es⋆e†. Intuitively this means that within an idempotent context that looks the same on the outside from semigroup. We introduce the notion of involution varieties of either direction, one can reverse factors. Bilateral semidirect languages and pseudovarieties of involution semigroups, and products with locally hermitian actions are called locally prove an Eilenberg correspondence between them in Section hermitian semidirect products. If V and W are pseudovarieties IV. In Section V, we conclude with some avenues for further of involution semigroups, the locally hermitian product of V inquiry. W and is the pseudovariety generated by all locally hermitian II. LANGUAGES OVER INVOLUTORY ALPHABETS semidirect products of involution semigroups from V and W. It is shown that a language is wLRTT if and only if it is A. Recognisable languages recognised by a locally hermitian semidirect product of an ape- Let A be a finite alphabet. Then, A+ denotes the set of riodic commutative semigroup and a locally trivial semigroup. nonempty finite words over A, and A∗ denotes the set of all This result can also be stated in terms of the corresponding finite words over A including the empty word ǫ. Extending this pseudovarieties of involution semigroups Acom⋆ and L1⋆. notation, A≤k denotes the subset of A∗ consisting of words Therefore, we have an algebraic characterisation of the class of length at most k, including ǫ. FO(A, min, max, N ). A semigroup S is a set together with an associative binary operation. Unless specified, the semigroup operation is written as x · y, for semigroup elements x, y; but we omit notating it Related work. The FO and MSO logics using the predicates whenever possible. If the operation has an identity, then S is N and bet were introduced in [8], [9]. Formulas of these logics called a monoid. All semigroups and monoids we consider are that do not use the constants min and max, are self-dual, finite, except when stated otherwise. For a finite alphabet A, i.e., replacing each predicate by its dual, for instance N(x, y) the set A+ forms a semigroup under concatenation, while the by N(y, x), results in an equivalent formula. Therefore, set A∗ is a monoid with the empty word ǫ as the identity. The MSO(A, N ) and MSO(A, bet) recognise precisely the class set S′ ⊆ S forms a subsemigroup of S if S′ is closed under of reversible regular languages — a language is reversible if it the semigroup operation. is invariant under taking the reverse of words in it. Similarly, Let (T, +) be a semigroup. A semigroup morphism from S FO(A, bet) recognises all aperiodic reversible languages. In to T is a map h: S → T that satisfies the equation h(xy)= [8], the question of deciding membership in these classes was h(x)+ h(y), for all x, y ∈ S. If S and T are monoids, then h studied. For the logics MSO(A, bet), FO(A, bet), the exist- is a monoid morphism if additionally h maps the identity of S ing decidable characterisations of MSO(A, <) and FO(A, <) to the identity of T . A semigroup T divides the semigroup S if immediately yield an answer. For the case of FO(A, N ), a T is the image of a subsemigroup of S under some morphism. partial answer was given; it was shown that if a language L is Let A be a finite alphabet. A language L ⊆ A+ is recog- definable in FO(A, N ), then its syntactic semigroup satisfies nized by a semigroup S if there is a morphism h: A+ → S the identity ex⋆e⋆ = exe⋆, in addition to those defining the and a subset P ⊆ S such that L = h−1(P ). The set P is called class LTT. the accepting set of L. The class of languages recognised by A different but related between predicate (namely a(x, y), finite semigroups coincide with the class of regular languages for a ∈ A, is true if there is an a-labelled position between po- [10], p. 160. sitions x and y) was introduced in [11]–[13]. They characterise The above definitions are easily adapted to the case of lan- the expressive power of two-variable first-order logic with the ∗ 2 guages over A . The definitions are assumed in the subsequent order relation (FO (<)) enriched with the between predicates sections. a(x, y) for a ∈ A, and show an algebraic characterisation of the resulting family of languages. B. Languages over Involutory Alphabets Since the class of involution semigroups contains many A finite involutory alphabet A = (A, †) is a finite alphabet important classes of semigroups such as inverse semigroups, A with a bijection †: A → A such that (a†)† = a for each there is substantial literature on varieties of involution semi- letter a ∈ A. The function † is extended to words over A as groups from a semigroup theoretic perspective (see [6] for a † † † survey). However we are not aware of a language theoretic w = an ...a1 , (2) + †† † if w = a1 ··· an ∈ A . Clearly, w = w. Moreover, (uv) = Clearly, if a language is recognised by an involution semi- v†u†, for all words u, v ∈ A+. Hence, the operation †: A+ → group, then it is recognised by a semigroup as well. The other A+ is an involution: a function that is its own inverse. If L ⊆ direction also holds, as we show next. A+ is a language, then L† denotes the set {w† | w ∈ L}. Let S = (S, †) be an involution semigroup and let T If † is the identity function on A, then A is called a be a semigroup. By T op, we denote the opposite of T — hermitian alphabet. In that case, † is the reverse operation the semigroup with the same set of elements but reversed u 7→ ur, for u ∈ A+, and L† is the set Lr, for L ⊆ A+. multiplication, i.e., a ·op b = b · a, for all a,b ∈ T . Assume Unless specified, a finite alphabet is taken to be a hermitian there is a semigroup morphism from S to T . Let h† be the alphabet, i.e. with the reverse operation as the involution. map from S to T op, given by, for each s ∈ S, ◮ Definition 3. An involution semigroup (also called a ⋆- h†(s)= h(s†) . semigroup) S = (S,⋆) is a semigroup S extended with an ◮ Lemma 1. If h: S → T is a morphism, then h† : S → T op operation ⋆: S → S such that for all elements a,b of S, is also a morphism. ⋆ ⋆ ⋆ ⋆ ⋆ (a ) = a , and (a · b) = b · a . Proof. For x, y ∈ S, S We say that S is the semigroup reduct of . The semigroup h†(xy)= h((xy)†) A+ with the operation †: A+ → A+, defined in (2), forms h y†x† an involution semigroup, denoted as A +. Similarly, A∗ with = ( ) † † † forms a monoid with an involution, denoted as A ∗. The = h(y )h(x ) (product in T ) + involution semigroup A is not a free involution semigroup = h†(x)h†(y) (product in T op) . in general; it is only so when the relation † is irreflexive on † op A ( [14], p. 172). Hence h : S → T is a semigroup morphism. ◭ A ◮ Example 1. Let T = {a,b,ab,ba} be the semigroup with Let = (A, †) be an involutory alphabet. Assume that + the multiplication L ⊆ A is recognised by the semigroup S with the morphism h and the accepting set P . Lemma 1 gives that h† : A+ → Sop aa = a, bb = b, aba = bab = ba. is also a morphism; by definition, h†(w†) = h(w), for all w ∈ A+. Hence, h†(L†) = h(L) = P ; hence, the semigroup Let ⋆ be the map a⋆ = b. The map extends to an involution Sop recognises L† with the morphism h†. on T , Next, we show that (T × T op, flip : (x, y) 7→ (y, x)) is an b⋆ = (a⋆)⋆ = a , (ab)⋆ = b⋆a⋆ = ab , (ba)⋆ = a⋆b⋆ = ba . involution semigroup. It suffices to show ◮ op However, the map a† = a,b† = b is not a well-defined Lemma 2. flip is an involution on T × T . involution on T ; Proof. Clearly, for (a,b) ∈ T × T op † † † † † † † † flip (ba) = a b = ab 6= ba = aba = a b a = (aba) = (ba) . (a,b)flip = (a,b) . It is not possible to associate an involution with every Let (a,b), (c, d) ∈ T × T op. Then, semigroup; for instance, the semigroup of right zeros, where flip flip the multiplication follows the identity xy = y for any elements ((a,b)(c, d)) = (ac,db) x, y, does not admit an involution if it has more than one = (db,ac) element. However, it is possible to obtain a ⋆-semigroup, from = (d, c)(b,a) a given semigroup, that shares similar properties, as we shall = (c, d)flip(a,b)flip . see later. Let S = (S,⋆) and T = (T, †) be involution semigroups. Hence flip is an involution on T × T op. ◭ A morphism of involution semigroups, also called an involu- ◮ Proposition 1. If a language is recognised by a semi- tory morphism, from S to T , denoted as h: S → T , is a group, then it is recognised by a ⋆-semigroup as well. semigroup morphism h: S → T such that h(a⋆)= h(a)†, for all a ∈ S. Proof. Let A = (A, †) be an involutory alphabet. Assume that L is recognised by S using the morphism h: A+ → S. ◮ Definition 4. Let A = (A, †) be an involutory alphabet. By Lemma 1, h† : A+ → Sop is a morphism. By Lemma 2, The involution semigroup S = (S,⋆) recognises the language (S × Sop, flip) is an involution semigroup. Then, the product L ⊆ A+ if there is a morphism of involution semigroups map g : w 7→ (h(w),h†(w)), for w ∈ A+, is a morphism from h: A + → S and a subset P ⊆ S such that L = h−1(P ). A+ to S × Sop. Since ◮ Example 2. Let A be the involutory alphabet {a,b} with g(w†) = (h(w†),h†(w†)) = (h†(w),h(w)) = (g(w))flip , the map a⋆ = b. Let T = {a,b,ab,ba} be the semigroup from Example 1. Then, T accepts the language L = a+b+ with the g is a morphism of involution semigroups from A + to morphism h(a)= a,h(b)= b and the accepting set {ab}. S × Sop. Finally, the proof is completed by observing that L is recognised by the involution semigroup S × Sop with the The actions l and r are compatible if for all t,t′ ∈ T , and accepting set {(h(w),h†(w)) | w ∈ L}. ◭ s ∈ S, it is the case that Therefore, languages recognised by involution semigroups (ts)t′ = t(st′) . (7) are precisely the class of recognisable languages. A pair of compatible actions (l, r) is called a bilateral Let S = (S,⋆) be an involution semigroup. If S′ ⊆ S, then action of T on S. Given a bilateral action (l, r) of T on S′⋆ denotes the set {s⋆ | s ∈ S′}. A subset S′ of S forms S, the bilateral semidirect product (also called the two-sided a sub-⋆-semigroup of S if S′ is closed under the semigroup semidirect product) S ∗∗T is the semigroup with the elements operation and the involution, i.e., if S′2 ⊆ S′ and S′⋆ = S′. {(s,t) | s ∈ S,t ∈ T } and with the operation If h: S → T is an involution semigroup morphism, then the T image of h is a sub-⋆-semigroup of . (s1,t1)(s2,t2) = (s1t2 + t1s2,t1t2) . (8) An element s ∈ S is hermitian if it is its own involution, i.e., Let S = (S,⋆) and T = (T, ⋄) be involution semigroups. if s⋆ = s.A ⋆-semigroup is hermitian-generated if it is gener- Let l, r be a bilateral action of T on S. Then, the pair l, r ated by its hermitian elements using the semigroup operation ( ) ( ) is an involutory action of T on S if for all t,t′ T,s S, and the involution. For convenience, we shorten hermitian- ∈ ∈ generated involution semigroup to hermitian semigroup. For (st)⋆ = t⋄s⋆ . (9) every hermitian alphabet A , the involution semigroup A + is ◮ T hermitian. Images of hermitian semigroups under involution Lemma 3. Assume = (T, ⋄) has an involutory action S ′ semigroup morphisms are hermitian as well. on = (S,⋆). Then, for all t,t ∈ T,s ∈ S, ⋆ ⋄ ⋆ ⋄ ◮ Example 3. In the involution semigroup T = (T,⋆) (t1st2) = t2s t1 . in Example 1, the set of hermitian elements is {ab,ba}. Since Proof. By taking ⋆ on both sides in (9), we get they don’t generate all the elements of the semigroup, T is not ⋄ ⋆ ⋆ a hermitian semigroup. But ({ab,ba},⋆) is a sub-⋆-semigroup st = (t s ) . (10) of T that is hermitian. Exchanging t and t⋄, and s and s⋆ in (10), we obtain ∗ For recognising languages that are subsets of A , the notion s⋆t⋄ = (ts)⋆ . (11) of recognisability by ⋆-semigroups is insufficient; one needs recognisability by ⋆-monoids. The definitions are straightfor- Finally, ward adaptations of the ones presented above. ⋆ ⋆ (t1st2) = ((t1s)t2) by (7) ⋄ ⋆ III.INVOLUTORY SEMIDIRECT PRODUCTS AND THE = t2(t1s) by (9) CHARACTERISATION OF FO ⋄ ⋆ ⋄ = t2s t1 . by (11) In this section, we first define the involutory semidirect ◭ product of two involution semigroups. Then, a particular case, called locally hermitian semidirect product, is defined. ◮ Definition 5. Assume T = (T, ⋄) has an involutory It is then shown that wLRTT languages are precisely the action on S = (S,⋆). Then, their involutory semidirect ones recognised by locally hermitian products of aperiodic product S ∗∗ T is the semigroup S ∗∗ T with the involution commutative ⋆-semigroups and locally trivial ⋆-semigroups. †: (s,t) 7→ (s⋆,t⋄) . (12) A. Bilateral Semidirect Products of Involution Semigroups ◮ Lemma 4. S ∗∗ T is an involution semigroup. Let S and T be finite semigroups. We denote the semigroup operation of S additively (by +) and of T multiplicatively. Proof. It suffices to show that † is an involution. By definition, † However, we don’t assume that S or T is commutative. A left (s,t)† = (s⋆⋆,t⋄⋄) = (s,t) . action l of T on S is a map l : T × S → S satisfying the following conditions. Denote the image of (t,s) under l as ts, Next, for all s1,s2 ∈ S,t1,t2 ∈ T , ′ for t ∈ T,s ∈ S. Then, for all s ∈ S and t,t ∈ T , † † ((s1,t1)(s2,t2)) = (s1t2 + t1s2,t1t2) by (8) ′ ′ ⋆ ⋄ t(s + s )= ts + ts , and, (3) = ((s1t2 + t1s2) , (t1t2) ) by (12) ′ ′ ⋆ ⋆ ⋄ ⋄ (tt )s = t(t s) . (4) = ((t1s2) + (s1t2) ,t2t1) ⋆ ⋄ ⋄ ⋆ ⋄ ⋄ Right actions of T on S are defined analogously; a right = (s2t1 + t2s1,t2t1) ⋆ ⋄ ⋆ ⋄ action r of T on S is a map r : S × T → S such that for all = (s2,t2) (s1,t1) s ∈ S and t,t′ ∈ T , † † = (s2,t2) (s1,t1) . ′ ′ (s + s )t = st + s t , and, (5) Hence † is an involution. ◭ s(tt′) = (st)t′ , (6) Next, we introduce a particular case of involutory semidirect where st denotes the image of (s,t) under the map r. product. An element e in a semigroup is an idempotent if e · e = e. The right action is analogous. In the case of involution semigroups, if e is an idempotent, xyˆ · first(w) if z is ǫ then involution of e is also an idempotent. xyzˆ ⊗ w = xyzˆ otherwise, ◮ Definition 6. Let S = (S,⋆) and T = (T, ⋄) be ( involution semigroups. Let (l, r) be an involutory action of T Let S′ be the semigroup formed by the powerset of U with on S . The action is locally hermitian if for each idempotent set union as the semigroup operation. For each element I ⊆ U e ∈ T , and each element s ∈ S, of S′, let Ir = {wr | w ∈ I}. The operation I 7→ Ir, for ′ ′ S ′ ese⋄ = es⋆e⋄ . (13) I ∈ S , is an involution on the semigroup S . Let = (S′, r). The action of T on U, extends to the elements of S ′ In other words, in the case of a locally hermitian action, pointwise. The elements A · Aˆ · A are called the zeros for the the elements of the form ese⋄ are hermitian (observe that, by action. Let ∼ be a congruence on S ′, given by, for I,J ∈ S′, ⋄ ⋆ ⋄⋄ ⋆ ⋄ ⋆ ⋄ Lemma 3, (ese ) = e s e = es e ). • x ∈ I if and only if x ∈ J, for every nonzero x ∈ U, S T T Let and be involution semigroups. If the action of and S r r is locally hermitian action on , then the semidirect product • x or x is in I if and only if x or x is in J, for each S T ∗∗ is called locally hermitian. zero x ∈ U. B. Characterisation of FO with Neighbour Let S be the quotient of S ′ with respect to ∼. It is easy S A semigroup S is aperiodic if there is an n N such that to see that for each idempotent e ∈ T , and each I ∈ , ∈ r r r an+1 = an, for all elements a in the semigroup. A semigroup e ⊗ I ⊗ e = e ⊗ I ⊗ e . Hence the actions are locally S T is commutative if the semigroup operation is commutative, i.e., hermitian. Finally L is accepted by ∗∗ with the morphism x·y = y·x, for all elements x and y. A semigroup S is locally h: a 7→ ({aˆ},a). trivial if ese = e, for each element s and each idempotent e in ◮ Theorem 2. Each wLRTT language is recognised by a S. The class of languages LTT has several characterisations in locally hermitian semidirect product of an aperiodic commuta- terms of various products of semigroups (see [1] for a detailed tive ⋆-semigroup and a locally trivial ⋆-semigroup. Conversely, discussion); the one relevant to us is, if L is recognised by a locally hermitian semidirect product S T S T ◮ Proposition 2. A language is in LTT if and only if it is ∗∗ , where is aperiodic and commutative and is recognised by a bilateral semidirect product of an aperiodic locally trivial, then L is wLRTT. commutative semigroup and a locally trivial semigroup. Proof. Assume L ⊆ A+, for an involutory alphabet A = Using the construction outlined in the proof of Proposition (A, id), is a wLRTT language. To prove the first claim, we S T S 1, it can be shown that a language is in LTT if and only define ⋆-semigroups and such that is aperiodic and T if it is recognised by an involutory semidirect product of commutative and is locally trivial, and show that their an aperiodic commutative involution semigroup and a locally locally hermitian semidirect product recognises L. We let r m trivial involution semigroup. k,m > 0 be such that L is a union of ≈2k+1-classes. Since r t r t Next, we characterise the class wLRTT. First, we give an the equivalence ≈d+1 refines the equivalence ≈d, for t,d > 0, example. there is no loss of generality. Let ⊙ be the operation on the set A≤2k \{ǫ} defined as ◮ Example 4. Consider the language L = (abc)+ over the alphabet A = {a,b,c}. We show that L is defined by a locally S T ′ ′ hermitian product of two ⋆-semigroups and . ′ t · t if |tt | < 2k, ≤2 t t Let T be the semigroup with the set A \{ǫ} and the ⊙ = ′ ′ (pref k(t · t ) · suff k(t · t ) otherwise, operation ⊙ given by, for x, y ∈ T where pref () and suff () denote the k-length prefix and x · y if |xy|≤ 2 k k x ⊙ y = suffix respectively. It is not difficult to see that the operation (first(xy) · last(xy) otherwise, ⊙ is associative. Hence, T = (A≤2k \{ǫ}, ⊙) is a semigroup. It is straight-forward to check that u v r vr ur and where first(w) and last(w) denote the first and last letter of a ( ⊙ ) = ⊙ r r T word w. Reverse operation is an involution on the semigroup (u ) = u, for the reverse operation r. Let = (T, r) be the involution semigroup with the reverse operation as the T . Let T = (T, r). Idempotents of T are precisely those involution. The idempotents of T are precisely the words of words of length 2. length k. Also, e t e e, for words t T and e A2k. Let Aˆ = {a,ˆ ˆb, cˆ}. Let U be the set A≤1 · Aˆ· A≤1. Elements 2 ⊙ ⊙ = ∈ ∈ Hence, T is locally trivial. of T act on words in U in the following way: for w ∈ T and xyzˆ ∈ U, where x, z ∈ A≤1 and yˆ ∈ Aˆ, we define the right Next, we set up some notation in preparation for defining S ˆ action ⊗ as . Let A denote the set {aˆ | a ∈ A} of anchored letters. A k-anchored word wˆ is a word from the set A≤k · Aˆ · A≤k. last(w) · yzˆ if x is ǫ Let W denote the set of all k-anchored words. We denote w ⊗ xyzˆ = k (xyzˆ otherwise, anchored words by w,ˆ uˆ etc. Let wˆ = uavˆ , where u, v ∈ A∗ and aˆ ∈ Aˆ. We write l(ˆw) The equivalence relation ≡m is a congruence relation on the m r m r and r(ˆw) to denote how far is the anchored letter from the involution semigroup M(Wk): If I ≡ J, then I ≡ J , left-end and right-end respectively; Hence, l(ˆw) = |u| and and also, if I ≡m J and K ≡m L, then I ∪ K ≡m J ∪ L, for r(ˆw)= |v|. Clearly, 0 ≤ l(ˆw), r(ˆw) ≤ k, for all wˆ ∈ Wk. all I,J,K,L ∈ M(Wk). m The semigroup T acts on the set Wk, on the left as well as We let S to be the quotient M(Wk)/≡ . Clearly, S is on the right, in the following manner. For each letter a ∈ A a finite semigroup. We denote the semigroup operation on and wˆ ∈ Wk, the left action a ⊗ wˆ ∈ Wk of a on wˆ is given S additively. The ⋆-semigroup S = (S, r) is aperiodic and by, commutative. wˆ if l(ˆw)= k, We can extend the involutory action ⊗ of T on Wk, to the a ⊗ wˆ = (14) ⋆-semigroup S pointwise, i.e., we let the left action u ⊗ I, (a · wˆ otherwise. for u ∈ T and I ∈ S, to be the multiset of anchored words Similarly the right action wˆ ⊗ a appends a to the right if with the support {u ⊗ wˆ | wˆ ∈ I} and multiplicity there are less than k positions to the right of the anchored position, otherwise it leaves wˆ intact. We extend these actions mu⊗I (ˆv)= mI (ˆw) , (18) w∈I,u⊗w v to all the elements of T by letting, for u ∈ T , a ∈ A and ˆ Xˆ=ˆ wˆ ∈ Wk, for vˆ ∈ Wk. Right action I ⊗u is defined dually. By Equations 15, 16, 17, the actions of T on S are involutory. It is easily (u ⊙ a) ⊗ wˆ = u ⊗ (a ⊗ wˆ) , and wˆ ⊗ (a ⊙ u)=(ˆw ⊗ a) ⊗ u . observed that, by definition of the action ⊗ and the congruence (15) ≡m, for each idempotent e ∈ T and each multiset I ∈ S, Clearly the left and right actions are compatible with each other, i.e., e ⊗ I ⊗ er = e ⊗ Ir ⊗ er . (19) (u ⊗ wˆ) ⊗ v = u ⊗ (ˆw ⊗ v) (16) Hence the action of T on S is locally hermitian. For for all wˆ ∈ Wk and u, v ∈ T . Also the actions are involutory, convenience, below we identify the anchored word wˆ and the with respect to the involution on T , since singleton multiset {wˆ}. Also, we will omit writing the action ⊗ explicitly. r r r (u ⊗ wˆ) =w ˆ ⊗ u . (17) Let S ∗∗ T be the locally hermitian semidirect product of S and T with the action . We define the ⋆-morphism h An anchored word is a zero for the action ⊗ if for all a ∈ A, ⊗ from A + to S ∗∗ T as a⊗wˆ =w ˆ⊗a =w ˆ. The zero elements of Wk are precisely the anchored words whose anchored letter is at distance k from h: a 7→ (â,a) ∈ S × T. each end. Next, we define the semigroup S.A multiset in the universe It is easy to verify that h is a morphism of involution U, where U is a set, is a collection of elements from U semigroups. that allows for multiple instances. For instance, the multiset Next, we show that the morphism h recognises the language r m ′ ′ {3, 3, 4}, in the universe N, contains two occurrences of 3 L; it suffices to show that if w ≈2k+1 w , then h(w)= h(w ). + ′ ′ ′ and one occurrence of 4. Given a multiset I and an element Assume w = a1 ··· an ∈ A and w = a1 ··· aℓ are two r m ′ ′ x ∈ I, the multiplicity of x in I, denoted as mI (x), is the words such that w ≈2k+1 w . If |w| < 2k+1 or |w | < 2k+1, number of times x occurs in I. The support of I, denoted as then clearly w = w′ and h(w) = h(w′). Therefore, assume Supp(I), is the set of all elements occurring in I. Given two that both words have length at least 2k +1. The image of w multisets I and J, the union of I and J, denoted as I ∪ J, under h is of the form is the multiset K with the support Supp(I) ∪ Supp(J) and h(w)=(â1,a1) ··· (ân,an) multiplicity mK (x)= mI (x)+ mJ (x), for all x ∈ Supp(K). Let M(U) denote the set of all non-empty multisets in the =(â1a2 ··· an + a1aˆ2 ··· an + ··· + a1 ··· an−1aˆn, universe U. The set M(U) is a semigroup with ∪ as the a1 ··· an) . semigroup operation. (20) Consider the semigroup M(Wk). Clearly, M(Wk) is in- Observe that if u, v ∈ T are such that |u|, |v|≥ k, then u⊗wˆ⊗ finite. For each multiset I in M W , we let Ir denote ( k) v = suff (u)⊗wˆ⊗pref (v), and u⊙v = pref (u)·suff (v), the multiset with support {xr | x ∈ I} and multiplicity k k k k r for each wˆ ∈ S. Since n ≥ 2k +1, we derive, mIr (x ) = mI (x), for each x ∈ I. The reverse operation is an involution on M(Wk). h(w)=(â1a2 ··· ak+1 + a1aˆ2 ··· ak+2 + ··· We define the following equivalence relation m on the ≡ + an−k ··· an−1aˆn, a1 ··· akan−k+1 ··· an) . m multisets in M(Wk); for I,J in M(Wk), I ≡ J if ′ ′ ′ m Similarly for w = a ··· a the image is of the form 1) mI (x)= mJ (x), for each non-zero element x of Wk, 1 ℓ and, h(w′) = (aˆ′ a′ ··· a′ + a′ aˆ′ ··· a′ + ··· r m r 1 2 k+1 1 2 k+2 2) mI (x)+ mI (x ) = mJ (x)+ mJ (x ), for each zero + a′ ··· a′ aˆ′ , a′ ··· a′ a′ ··· a′ ) element x of Wk. ℓ−k ℓ−1 ℓ 1 k ℓ−k+1 ℓ ′ ′ ′ ′ A + ′ Since w and w have a common prefix and suffix of length Similarly, let w = a1 ··· am ∈ . Assuming that h(ai)= ′ ′ ′ 2k, their images in T are identical. Hence, all it remains to (si,ti), the image of w under h is conclude the claim is to show that the summations ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ h(w ) = (s1t2 ··· tm + t1s2t3 ··· tm + ··· + t1 ··· tm−1sm, L =a ˆ a ··· ak + a aˆ ··· ak + ··· + an−k ··· an− aˆn ′ ′ 1 2 +1 1 2 +2 1 t1 ··· tm) . ˆ′ ′ ′ ′ ˆ′ ′ ′ ′ ˆ′ (24) R = a1a2 ··· ak+1 + a1a2 ··· ak+2 + ··· + aℓ−k ··· aℓ−1aℓ r t ′ have the same value in S. Observe that, since S is aperiodic Since m,n ≥ 4k +1 and w ≈4k+1 w , by definition, the and commutative, we can freely commute the terms in L and words w and w′ have the same prefix and suffix of length ′ ′ R. 4k. Using Equation 22, we conclude that t1 ··· tn = t1 ··· tm. The zero elements in the summation L are precisely the Therefore, it only remains to show that the summations terms of the form L = s1t2 ··· tn + t1s2t3 ··· tn + ··· + t1 ··· tn−1sn ai−k ··· ai−1aîai+1 ··· ai+k , ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ R = s1t2 ··· tm + t1s2t3 ··· tm + ··· + t1 ··· tm−1sm where k 0, such that st = st+1 for all elements s ∈ S. i i r i=1 i=1 Let w, w′ ∈ A + be two words such that w ≈t w′. We X X 4k+1 are identical. Similarly, since the words have the same suffix claim that w ∈ L if and only if w′ ∈ L; proving the claim implies the lemma. We assume that w and w′ are of length at of length 4k, the partial sums least 4k +1, otherwise w = w′, and the claim is immediate. n m + Li and Ri Let w = a1 ··· an ∈ A be a word. Let h(ai) = (si,ti). i n− k i m− k The image of w under h is = X2 +1 = X2 +1 are also identical. Thus, all it remains is to show is the equality h(w) = (s ,t ) ··· (s ,t ) 1 1 n n of the following partial sums: = (s1t2 ··· tn + t1s2t3 ··· tn + ··· + t1 ··· tn−1sn, n−2k m−2k t1 ··· tn) . Lf = Li and Rf = Ri . (23) i k i k X=2 X=2 The terms occurring in Lf and Rf are called special. finite/infinite words, finite trees etc. can be seen as instances of ′ ′ Let x = t1 ··· tk = t1 ··· tk and y = tn−k+1 ··· tn = this unified result. In this framework, the involution operation ′ ′ tm−k+1 ··· tm. Observe that, for each special term Li, is a monad that satisfies a certain distributive law. One advantage of this approach is that the variety theorem for L x t t s t t y . i = · ( i−k ··· i−1) i( i+1 ··· i+1+k) · involutary languages can be deduced from the general case.

Similarly, for each special term Ri, But it also requires the significantly more complex language of categories. For the sake of simplicity we do the variety ′ ′ ′ ′ ′ Ri = x · (ti−k ··· ti−1) si(ti+1 ··· ti+1+k) · y . theorem in the usual way. Before proceeding, we make an observation about special A. Syntactic Involution Semigroups terms. Let e and e⋄ be two idempotents in T . Let p,q,u,v ∈ T + be such that |p|, |q|, |u|, |v|≥ k, and let s ∈ S, then, The syntactic congruence of a language L ⊆ A+, denoted + as ∼L, is the congruence relation on A , defined as pqsuv = pe(qsu)e⋄v (usingEquation22) (27) = pe(qsu)⋆e⋄v (∵ the actions are locally hermitian) x ∼L y := uxv ∈ L if and only if uyv ∈ L, (33) (28) for all words u, v ∈ A∗. ⋄ ⋆ ⋄ ⋄ ∵ + = peu s q e v ( the actions are involutory) (29) The quotient semigroup A /∼L, denoted as S(L), is called ⋄ ⋆ ⋄ = pu s q v . (usingEquation22) (30) the syntactic semigroup of L. Let [u]L denote the equivalence + class of u ∈ A under the congruence ∼L. Then, the surjective Applying the above observation and Equation 21 to the term + + morphism ϕL : u 7→ [u]L, for u ∈ A , from A to S(L), Li implies that recognises L with the accepting set ϕL(L). The morphism x·(ti−k ··· ti−1) si(ti+1 ··· ti+1+k) · y ϕL is called the syntactic morphism of L. ⋄ ⋄ ⋆ ⋄ ⋄ It is possible to lift the notion of syntactic semigroup to the = x · (ti+1+k ··· ti+1) si (ti−1 ··· ti−k) · y (31) case of recognition by involution semigroups. = x · (ti+1+k ··· ti+1) si(ti−1 ··· ti−k) · y . ◮ Definition 7. Let A = (A, †) be an involutory alphabet. Similarly for Ri, + The syntactic ⋆-congruence of L ⊆ A is the relation ≈L on ′ ′ ′ ′ ′ + x·(ti−k ··· ti−1) si(ti+1 ··· ti+1+k) · y A , given by ′⋄ ′⋄ ′ ⋆ ′⋄ ′⋄ = x · (ti+1+k ··· ti+1) si (ti−1 ··· ti−k) · y (32) x ≈ y := x ∼ y and x ∼ † y . ′ ′ ′ ′ ′ L L L = x · (ti+1+k ··· ti+1) si(ti−1 ··· ti−k) · y Since ≈ is an intersection of two congruence relations on We are ready to conclude the claim L = R . Factors of L f f A+, it is a congruence on A+. Hence, A+/≈ is a semigroup. w, of length 2k +1, and centred at positions i, for 2k ≤ i< L Let [[w]] denote the equivalence class of w ∈ A+ under n − 2k, are called special. Similarly, factors of w′, of length L the congruence ≈ . Firstly, we observe 2k +1, and centred at positions i, for 2k ≤ i

† + semigroups that is closed under ﬁnite direct products, sub- ◮ Lemma 7. The map α:[w] 7→ [w ] † , for w ∈ A , is L L semigroups and quotients. an isomorphism from S(L)op to S(L†). If S = (S,⋆) and T = (T, †) are two ⋆-semigroups then Proof. By Lemma 5, the map is well-deﬁned. Clearly, α is the direct product of S and T , denoted S × T , is the surjective. By the same lemma, it is injective. We verify that semigroup S × T with the involution (x, y) → (x⋆,y†).

op α([u]L[v]L)= α([vu]L) (Product in S(L) ) ◮ Definition 9. A class of finite involution semigroups V is † an involution pseudovariety if it is closed under taking finite = [(vu) ]L† † † direct products, sub-⋆-semigroups and quotients. = [u v ]L† ⋆ † † If S is a pseudovariety of finite semigroups, then S = [u ] † [v ] † L L is the pseudovariety of finite involution semigroups whose Hence, α is an isomorphism. ◭ semigroup reducts are in S. ◮ Example 5. 1) L1 is the pseudovariety of all finite ◮ Theorem 3. The correspondences V 7→ V and V 7→ V semigroups S such that S is locally trivial, i.e., for define mutually inverse bijective correspondences between each idempotent e ∈ S and element x ∈ S, exe = e. pseudovarieties of involution semigroups and ⋆-varieties of Hence, L1⋆ is the set of all involution semigroups whose languages. semigroup reducts are in L1. Rest of this subsection is devoted to the proof of the above 2) Acom⋆ is the pseudovariety of all finite involution theorem. The following lemma is easy to verify. semigroups (S,⋆) such that S is commutative and aperiodic. Since S is commutative, the identity function id ◮ Lemma 8. The correspondences are well defined. i.e. is always an involution on S. 1) If V is a ⋆-variety of languages and V 7→ V, then V is We let Reg denote the family of all regular languages a pseudovariety of involution semigroups. A V over the alphabet A. A variety of recognisable languages is 2) If is a pseudovariety of involution semigroups and V a function V that maps each finite alphabet A to a subset of 7→ V, then V is a ⋆-variety of languages. RegA such that the following conditions hold: Next we show that the maps are mutually inverse bijections. 1) For each alphabet A, the family of languages V(A) is The proof follows a standard recipe; the one outlined here closed under union, intersection and complement, i.e., closely follows the proof of the Variety theorem for finite V(A) is a Boolean algebra. semigroups given in [16]. 2) For each L ∈V(A) and each letter a ∈ A, the quotients ◮ Proposition 5. Every involution semigroup S divides a−1L and La−1 are in V(A). a finite direct product of syntactic ⋆-semigroups of languages 3) For each morphism h: A+ → B+, if L ∈ V(B), then recognised by S . h−1(L) ∈V(A). Proof. Let S S,⋆ be an involution semigroup. Let A S ◮ Example 6. 1) L1 is the variety of languages where = ( ) ⊆ be such that A generates S and be the involution ⋆ restricted L1(A) is the set of all languages L ⊆ A+ that are † A A + S finite boolean combinations of languages of the form to A. Let = (A, †). Let h: → be the surjective ⋆-morphism that maps each sequence of elements to their uA∗ or A∗u, for some u ∈ A+. The language variety product in S. For s S, let L denote the language recognised L1 corresponds to the pseudovariety L1 [16]. ∈ s by the set s , in other words L h−1 s . Let S S ,⋆ 2) Let L(a, k) ⊆ A+ be the set of all words in which letter { } s = ( ) s = ( s s) be the syntactic ⋆-semigroup of L and let χ be the syntactic a ∈ A appears exactly k times. Acom is the variety s s ⋆-morphism. where Acom(A) is the set of all languages L ⊆ A+ S that are finite boolean combinations of languages of the Since recognises the languages Ls, there exists a sur- S S form L(a, k) [16], [22]. jective morphism ψs from to s, by Proposition 4. Let Πs∈S Ss denote the direct product of the syntactic ⋆- ◮ Definition 10. A involution variety of languages, also semigroups Ss corresponding to each element of S . Let called a ⋆-variety of languages, is a function that maps each V s∈S χs denote the product map u 7→ (χs(u))s∈S, for A + involutory alphabet = (A, †) to a family of recognisable u ∈ A . Similarly, ψ = ψs denotes the product map Q s∈S languages over the alphabet A such that the following holds. from S to Ss. As product of involutory morphisms s∈S Q 1) For each A , the set V(A ) is a Boolean algebra. these maps are involutory morphisms. Q 2) For each L ∈V(A ) and each letter a ∈ A, the quotients The various maps and semigroups are detailed in Figure a−1L and La−1 are in V(A ). IV-C. 3) Let h: (A+, †) → (B+,⋆) be a morphism. Then, if L ∈ We claim that ψ is injective. Assume ψ(s)= ψ(s′) for some −1 ′ ′ V(B), then h (L) ∈V(A ). s,s ∈ S. Then in particular ψs(s)= ψs(s ). This implies that 4) If L ∈V(A ), then L† is in V(A ). for all x, y ∈ S1, xsy = s ⇔ xs′y = s. In particular when x = y = 1, this implies s = s′. Hence, ψ is injective. Therefore C. Eilenberg-type correspondence S S ◭ is isomorphic to a sub-⋆-semigroup of s∈S s. We define a correspondence between ⋆-varieties of lan- ◮ V Q guages and pseudovarieties of finite involution semigroups. We Lemma 9. The map 7→ V is injective. write V, W,... for pseudovarieties of finite involution semi- Proof. Assume that V 7→ V and W 7→ W. It suffices to show groups and V, W,... for involution varieties of languages. that if V⊆W, then V ⊆ W. If so, then With each involution variety of languages V, we associate a pseudovariety of involution semigroups V, denoted as V 7→ V, V = W =⇒ V⊆W and W⊆V where V is the pseudovariety generated by the syntactic ⋆- =⇒ V ⊆ W and W ⊆ V semigroups of languages in V. Likewise, we associate an =⇒ V = W , involution variety of languages V with every pseudovariety of involution semigroups V, denoted as V 7→ V, where V maps and the map is injective. Next, we prove this. each involutory finite alphabet (A, †) to all the languages over Assume that V ⊆W. Let S = (S,⋆) be a ⋆-semigroup A recognised by ⋆-semigroups in V. in V. We show that S is in W as well. Let K be the h Since A is closed under union, it suffices to establish (A+, †) / S V( ) ③ that for every s ∈ P , one has that ϕ−1(s) belongs to V(A ). ③③ ③③ Let s = (s ,...,s ), then s = π−1(s ). Hence ③③ 1 n 1≤i≤n i i ③③ Qs S χs ③ ∈ ③ −1 −1 −1T −1 ③③ ψ=Qs∈S ψs ϕ (s)= ϕ π (s ) = ϕ (s ) . ③③ i i i i ③③ 1≤i≤n 1≤i≤n ③| ③ \ \ S s∈S s As V(A ) is closed under union and intersection, it is enough to show that ϕ−1 s is in A for each i n. (a) VariousQ maps in the proof of Proposi- i i V( ) 1 ≤ ≤ tion 5. Since ϕ = χ ◦ ψ , one has ϕ−1(s ) = ψ−1(χ−1(s )). Now i i i i i i i i A + ψi / A + since V is a involution variety of languages, it suffices to prove ❋❋ i that χ−1 s is in A , that results from the Lemma 11. ❋❋ i i V( i) ❋❋ ◭ ❋❋ ❋ ϕi ϕ ❋❋ χi ❋❋ ◮ ❋❋ Lemma 11. Let V be an involution variety of languages ❋❋ A + S ❋ and let χ: → (L) be the syntactic ⋆-morphism of a ❋# − πi / language L of V(A ). Then for every x ∈ S (L), χ 1(x) T ⊆ S / S (Li) belongs to V(A ). (b) Diagram in Lemma 10 S Fig. 2. Proof. Assume (L) = (S,⋆). Fix an element x ∈ S. Let P = χ(L). Then L = χ−1(P ). Setting E = {(s, t) ∈ S2 | sxt ∈ P }, we claim that set of all languages recognised by S . The languages in K −1 −1 −1 −1 are in V and hence in W also. This means there exist ⋆- {x} = s Pt \ s Pt c semigroups recognising languages in K in the pseudovariety (s,t\)∈E (s,[ t)∈E W, and hence the syntactic ⋆-semigroups of languages in K Let R be the right-hand side of the above equation. It is clear W are in , by Proposition 4. Therefore, their product is also that x belongs to R. Conversely, let u be an element of R. W S S in . Since divides this product by Proposition 5, is That means if sxt ∈ P , then sut ∈ P and if sxt 6∈ P , then in W. Since S was arbitrary, this shows that V W. ◭ ⊆ sut 6∈ P . It follows that u ∼L x and thus u = x, which proves the claim. ◮ Lemma 10. If V 7→ V 7→ W then V = W. Since χ−1 commutes with Boolean operations and quo- Proof. Assume that V 7→ V 7→ W. Firstly, if L ∈V(A ), for tients, χ−1(x) is some A = (A, †), then one has S (L) ∈ V by definition of the map V 7→ V. Hence, L ∈W(A ), by definition of the map −1 −1 −1 −1 −1 −1 −1 V 7→ W. Therefore, V(A ) ⊆ W(A ), for every involutory χ (x)= χ s Pt \ χ s Pt A (s,t)∈E (s, t)∈Ec alphabet . \ [ A A Next we prove the inclusion W( ) ⊆ V( ). Let L ∈ which is a Boolean combination of quotients of L and hence A S V V W( ). Then, (L) ∈ , and since is the pseudova- belongs to V(A +). ◭ riety generated by syntactic ⋆-semigroups of involutory languages in V, there exists finitely many of languages L1 ∈ From Lemma 9 and Lemma 10 we conclude Theorem 3. V(A1),...,Ln ∈ V(An), where n > 0 and Ai = (Ai, †i), such that S (L) divides S (L1) ×···× S (Ln). Let S = D. Bilateral Semidirect Products of Involution Pseudovarieties S L S L . Since S L divides S , it is a quotient ( 1)×···× ( n) ( ) If V and W are pseudovarieties of finite semigroups, of a sub-⋆-semigroup T T,⋆ of S . = ( ) then the bilateral semidirect product of the pseudovarieties T Observe that recognises L. Therefore there exists a is the pseudovariety generated by all the bilateral semidirect surjective ⋆-morphism ϕ A + T and a subset P of T : → products S ∗∗ T with S ∈ V, T ∈ W. Likewise, if V and −1 S S th such that L = ϕ (P ). Let πi : → (Li) be the i W are pseudovarieties of involution semigroups, then their projection defined by πi(s1,...,sn) = si. Put ϕi = πi ◦ ϕ bilateral semidirect product is the pseudovariety generated by and let χ A + S L be the syntactic morphism of L . i : i → ( i) i all involutory semidirect products of the form S ∗∗T , where Since χ is surjective, there exists a morphism of semigroups i S ∈ V and T ∈ W. A + A + ψi : → i such that ϕi = χi ◦ ψi. We can summarise the situation in a diagram in Figure 2. ◮ Definition 11. The locally-hermitian semidirect product Next we show that L ∈ V(A ) that proves the claim of the pseudovarieties of involution semigroups V and W is W(A ) ⊆V(A ). Firstly, the pseudovariety generated by all locally hermitian products S ∗∗ T , where S ∈ V and T ∈ W. L = ϕ−1(P )= ϕ−1(s) From Proposition 2, we deduce s∈P [ ◮ Proposition 6. The involution variety LTT corresponds [5] Janusz A. Brzozowski and Imre Simon. Characterizations of locally to the bilateral semidirect product of the pseudovarieties testable events. In Proceedings of the 12th Annual Symposium on Acom∗ L1⋆ Switching and Automata Theory (Swat 1971), SWAT ’71, pages 166– and . 176, 1971. [6] Siniˇsa Crvenkovićand Igor Dolinka. Varieties of involution semigroups Definition 2 and Theorem 2 can be generalized to the case of and involution semirings: a survey. 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