:: Basic Notation of Universal Algebra
:: by Jaros{\l}aw Kotowicz, Beata Madras and Ma{\l}gorzata Korolkiewicz
::
:: Received December 29, 1992
:: Copyright (c) 1992-2021 Association of Mizar Users


definition
attr c1 is strict ;
struct UAStr -> 1-sorted ;
aggr UAStr(# carrier, charact #) -> UAStr ;
sel charact c1 -> PFuncFinSequence of the carrier of c1;
end;

registration
cluster non empty strict for UAStr ;
existence
ex b1 being UAStr st
( not b1 is empty & b1 is strict )
proof end;
end;

registration
let D be non empty set ;
let c be PFuncFinSequence of D;
cluster UAStr(# D,c #) -> non empty ;
coherence
not UAStr(# D,c #) is empty ;
end;

definition
let IT be UAStr ;
attr IT is partial means :Def1: :: UNIALG_1:def 1
 the charact of IT is homogeneous ;
attr IT is quasi_total means :Def2: :: UNIALG_1:def 2
 the charact of IT is quasi_total ;
attr IT is non-empty means :Def3: :: UNIALG_1:def 3
( the charact of IT <> {} & the charact of IT is non-empty );
end;

:: deftheorem Def1 defines partial UNIALG_1:def 1 :
for IT being UAStr holds
( IT is partial iff the charact of IT is homogeneous );

:: deftheorem Def2 defines quasi_total UNIALG_1:def 2 :
for IT being UAStr holds
( IT is quasi_total iff the charact of IT is quasi_total );

:: deftheorem Def3 defines non-empty UNIALG_1:def 3 :
for IT being UAStr holds
( IT is non-empty iff ( the charact of IT <> {} & the charact of IT is non-empty ) );

registration
cluster non empty strict partial quasi_total non-empty for UAStr ;
existence
ex b1 being UAStr st
( b1 is quasi_total & b1 is partial & b1 is non-empty & b1 is strict & not b1 is empty )
proof end;
end;

registration
let U1 be partial UAStr ;
cluster the charact of U1 -> homogeneous ;
coherence
the charact of U1 is homogeneous by Def1;
end;

registration
let U1 be quasi_total UAStr ;
cluster the charact of U1 -> quasi_total ;
coherence
the charact of U1 is quasi_total by Def2;
end;

registration
let U1 be non-empty UAStr ;
cluster the charact of U1 -> non-empty non empty ;
coherence
( the charact of U1 is non-empty & not the charact of U1 is empty ) by Def3;
end;

definition
mode Universal_Algebra is non empty partial quasi_total non-empty UAStr ;
end;

theorem :: UNIALG_1:1
for n being Nat
for U1 being non empty partial non-empty UAStr st n in dom the charact of U1 holds
the charact of U1 . n is PartFunc of ( the carrier of U1 *), the carrier of U1 ;

definition
let U1 be non empty partial non-empty UAStr ;
func signature U1 -> FinSequence of NAT means :: UNIALG_1:def 4
( len it = len the charact of U1 & ( for n being Nat st n in dom it holds
for h being non empty homogeneous PartFunc of ( the carrier of U1 *), the carrier of U1 st h = the charact of U1 . n holds
it . n = arity h ) );
existence
ex b1 being FinSequence of NAT st
( len b1 = len the charact of U1 & ( for n being Nat st n in dom b1 holds
for h being non empty homogeneous PartFunc of ( the carrier of U1 *), the carrier of U1 st h = the charact of U1 . n holds
b1 . n = arity h ) )
proof end;
uniqueness
for b1, b2 being FinSequence of NAT st len b1 = len the charact of U1 & ( for n being Nat st n in dom b1 holds
for h being non empty homogeneous PartFunc of ( the carrier of U1 *), the carrier of U1 st h = the charact of U1 . n holds
b1 . n = arity h ) & len b2 = len the charact of U1 & ( for n being Nat st n in dom b2 holds
for h being non empty homogeneous PartFunc of ( the carrier of U1 *), the carrier of U1 st h = the charact of U1 . n holds
b2 . n = arity h ) holds
b1 = b2
proof end;
end;

:: deftheorem defines signature UNIALG_1:def 4 :
for U1 being non empty partial non-empty UAStr
for b2 being FinSequence of NAT holds
( b2 = signature U1 iff ( len b2 = len the charact of U1 & ( for n being Nat st n in dom b2 holds
for h being non empty homogeneous PartFunc of ( the carrier of U1 *), the carrier of U1 st h = the charact of U1 . n holds
b2 . n = arity h ) ) );

:: from MSSUBLAT, 2007.05.13, A.T.
registration
let U0 be Universal_Algebra;
cluster the charact of U0 -> Function-yielding ;
coherence
the charact of U0 is Function-yielding ;
end;