Multiclones on two-element set

Transcription

1 Irkutsk State Teacher Training University Grant RFBR «Mal tsev Meeting» Novosibirsk, august, 29

2 Clones partial clones hyperclones multiclones Let A nonempty set, n. n-ary operation f : A n A; n-ary partial operation f : A n A ; n-ary hyperoperation f : A n 2 A \ ; n-ary multioperation f : A n 2 A.

3 Clones partial clones hyperclones multiclones Composition h(x,..., x n ) = f (f (x,...x n ),..., f m (x,...x n ))

4 Clones partial clones hyperclones multiclones Composition h(x,..., x n ) = f (f (x,...x n ),..., f m (x,...x n )) composition operations h(a,...a n ) = f (f (a,...a n ),..., f m (a,...a n ))

5 Clones partial clones hyperclones multiclones Composition h(x,..., x n ) = f (f (x,...x n ),..., f m (x,...x n )) composition operations h(a,...a n ) = f (f (a,...a n ),..., f m (a,...a n )) composition partial operations {, if fi (a h(a,..., a n ) =,..., a n ) = for i {,...n}; f (f (a,...a n ),..., f m (a,...a n ))

6 Clones partial clones hyperclones multiclones Composition h(x,..., x n ) = f (f (x,...x n ),..., f m (x,...x n )) composition operations h(a,...a n ) = f (f (a,...a n ),..., f m (a,...a n )) composition partial operations {, if fi (a h(a,..., a n ) =,..., a n ) = for i {,...n}; f (f (a,...a n ),..., f m (a,...a n )) composition hyperoperations h(a,..., a n ) = f (b,..., b m ), b i f i (a,..., a n )

7 Clones partial clones hyperclones multiclones Composition h(x,..., x n ) = f (f (x,...x n ),..., f m (x,...x n )) composition operations h(a,...a n ) = f (f (a,...a n ),..., f m (a,...a n )) composition partial operations {, if fi (a h(a,..., a n ) =,..., a n ) = for i {,...n}; f (f (a,...a n ),..., f m (a,...a n )) composition hyperoperations h(a,..., a n ) = f (b,..., b m ), b i f i (a,..., a n ) composition multioperations {, if fi (a h(a,..., a n ) =,..., a n ) = for i {,...n}; f (b,..., b m ), b i f i (a,..., a n )

8 Clones partial clones hyperclones multiclones Projections πi n : (x,...x n ) x i Hyperprojections ei n : (x,...x n ) {x i } A set of (partial) operations is a (partial) clone if it containts all projections and is closed with respect to composition. A set of (multi) hyperoperations is a (multi) hyperclone if it containts all hyperprojections and is closed with respect to composition.

9 Clones A is a rank of (partial, hyper, multi) clone. P 2 clone of all operations on A : A = 2; P 2 partial clone of all partial operations on A : A = 2; P2 hyperclone of all hyperoperations on A : A = 2; P2 multiclone of all multioperations on A : A = 2. Clones of rank 2 P 2 P2 P2 ; P 2 P2 P 2 Maximal subclones 5: T, T, S, M, L (Post E., 92). All subclones have described (Post E., 94);

10 Partial clones Partial clones of rank 2 Maximal subclones 8 (Wang Xianghao, 963; Freivald R.V., 966). Subclones have partly described (Alekseyev V.B., Voronenko A.A., 994; Strauch B., 997; Lau D., 26); A L(A) P 2 3 T a (a {, }) 6 M 6 S 6 T T 3 T a M(a {, }) 5 T T M Between L and P2 is continuum of partial clones (Alekseyev V.B., Voronenko A.A., 994.)

11 Hyperclones and multiclones of rank 2 There are 9 maximal hyperclones (Тарасов В.В., 976) There are 3 minimal hyperclones (Pantović J., Vojvodić G., 24); Lattice of multiclones, keeped clone of all operation, contains 6 elements (Doroslovački R, Pantović J., Vojvodić G., 25;, Peryazev N.A., 25)

12 Comlete sets Set of multioperations R = P {f }, where P complete set in the class P 2, f operation, such that f ( α) = and f ( β) = {, }, is complete; Set of multioperations R = U {()}, where U the set of unary operations, is complete.

13 Maximal multiclones of rank 2 There are 5 maximal multiclones. Coding: =, {} =, {} =, {, } =. K = {f (x,..., x n ) f или f (,..., ) {, }}. K 2 = {f (x,..., x n ) f или f (,..., ) {, }}. K 3 = {f (x,..., x n ) f (,..., ) {, }}. K 4 = {f (x,..., x n ) f (,..., ) {, }}. K 5 = P2 { }. K 6 = P2. K 7 = {f {f (,..., ), f (,..., )} [e].

14 Maximal multiclones of rank 2 Multioperation f (x,..., x n ) preserves the relation R B m (α,..., α m ),..., (α n,..., α mn ) R (f (α,..., α n ),..., f (α m,..., α mn )) R. e(x) = () preserves a relation at F = {,,, }. g = (), R = {( ), ( )}.

15 Maximal multiclones of rank 2 Multioperation f (x,..., x n ) preserves the relation R B m (α,..., α m ),..., (α n,..., α mn ) R (f (α,..., α n ),..., f (α m,..., α mn )) R. If g = f (f,..., f m ), then for any (γ,..., γ n ) ({,, }) n g(γ,..., γ n ) = f (f (γ,..., γ n ),..., f m (γ,..., γ n )).

16 Maximal multiclones of rank 2 Multioperation f (x,..., x n ) preserves the relation R B m (α,..., α m ),..., (α n,..., α mn ) R (f (α,..., α n ),..., f (α m,..., α mn )) R. If g = f (f,..., f m ), then for any (γ,..., γ n ) ({,,, }) n g(γ,..., γ n ) f (f (γ,..., γ n ),..., f m (γ,..., γ n )). Let f (x) = g(x, x), then f () = {f (), f ()} = {g(, ), g(, )} g(, ) = {g(, ), g(, ), g(, ), g(, )}

17 Maximal multiclones of rank 2 K 8 class of operations, preserved a relation ( ) R 8 =. K 9 class of operations, preserved a relation R 9 ( ) R 9 =.

18 Maximal multiclones of rank 2 K class of operations, preserved a relation R ( ) R =. K class of operations, preserved a relation R ( ) R =.

19 Maximal multiclones of rank 2 K 2 class of operations, preserved a relation R 2. R 2 does not contain. K 3 class of operations, preserved a relation R 3. R 3 does not contain.

20 Maximal multiclones of rank 2 K 4 class of operations, preserved a relation R 4. R 4 does not contain (α, β, γ ) α β γ and does not contain (δ δ 2 δ 3 δ 4 ) such that δ i (i {, 2, 3, 4}) and among δ, δ 2, δ 3, δ 4 α E and β = are meeted.

21 Maximal multiclones of rank 2 K 5 class of operations, preserved a relation R 5, does not contain (α, β, δ, ν {, }, µ {,, }) α β δ µ µ µ µ ν ν ν ν and dual.

22 Maximal multiclones of rank 2 K 5 class of operations, preserved a relation R 5, does not contain (α, β, δ, ν {, }, µ {,, }) α β δ µ µ µ µ ν ν ν ν and dual. The criteria of completeness Set of multioperations is complete if and only if it is not contained completelly in any classes K K 5.