In particular, the concept of non-deterministic algebras was in- troduced in Computer Science in order to deal with nondeterminism

Ana Claudia de Jesus Golzio Marcelo Esteban Coniglio UNICAMP

1

ǯȱћѡџќёѢѐѡіќћ

Non-determinism was considered in Computer Science since its beginnings: from non-deterministic Turing machines to models of concurrency, event structures and Petri nets, as well as for variants of process languages and of l-calculus, the use of multifunctions instead of ordinary functions (asigning to each element of the domain a set of possible choices, instead of a single value) has revealed to be a ex- tremely useful conceptual tool. Indeed, there is a need for abstraction when modelling computational procedures, by disregarding irrelevant information. Being so, instead of considering all the dependencies on all the possible parameters, they can be represented by (nondetermi- nistic) choices.

In particular, the concept of non-deterministic algebras was in- troduced in Computer Science in order to deal with nondeterminism.

Thus, for instance, non-deterministic algebras were proposed as an

Š•Ž›—Š’ŸŽȱ˜ȱŽę—Žȱ̕ȬȬ›ŽŽȬ›ŽŒ˜—’£Ž›œǰȱ ‘’Œ‘ȱŠ›ŽȱŽœ’—Žȱ˜ȱ›Ž-

Œ˜—’£ŽȱŽ›–œȱ›˜–ȱ‘Žȱ›ŽŽȱŠ•Ž‹›ŠȱŽ—Ž›ŠŽȱ‹¢ȱŠȱœ’—Šž›Žȱ̕ȱ›˜–ȱ

ŠȱœŽȱȱ˜ȱŽ—Ž›Š˜›œȱǻŒǯȱǽŗŘǾǼǯȱ—ȱ’—Ž›Žœ’—ȱ–˜—˜›Š™‘ȱ˜ȱȱ—˜—ȬŽ-

Carvalho, M.; Braida, C.; Salles, J. C.; Coniglio, M. E. ’•˜œ˜ęŠȱŠȱ’—žŠŽ–ȱŽȱŠȱà’ŒŠǯ

˜•Ž³¨˜ȱȱ—Œ˜—›˜ȱDZȱǰȱ™ǯȱřŘŝȬřŚŜǰȱŘŖŗśǯ

řŘŞ

terminism in Computer Science from an algebraic perspective can be

˜ž—ȱ’—ȱǽŗşǾǯ

In the realm of Logic, non-determinism was considered mainly as a tool for obtaining alternative semantics. Non-deterministic matri- ces constitute a good example of this alternative approach.

The non-deterministic matrices (Nmatrices, for short), introdu-

ŒŽȱ’—ȱǽŘǾǰȱǽřǾȱŠ—ȱǽŗǾǰȱŠ›ŽȱŠȱŽ—Ž›Š•’£Š’˜—ȱ˜ȱ‘ŽȱžœžŠ•ȱŒ˜—ŒŽ™ȱ˜ȱ•˜’Œȱ matrix^ŗȱŠ—ȱ‘Žȱ–Š’—ȱŽŠž›Žȱ˜ȱ‘’œȱŽ—Ž›Š•’£Š’˜—ȱ’œȱ‘Šȱ‘ŽȱŸŠ•žŽȱ‘Šȱ a valuation assigns to a complex formula can be chosen non-determi- nistically from a non-empty set options. That is, Nmatrices are based on non-deterministic algebras, in contrast with the usual logical matri- ces which are based on standard algebras.

Š—¢ȱ™›˜™˜œ’’˜—Š•ȱ•˜’ŒȱŒŠ—ȱ‹ŽȱœŽ–Š—’ŒŠ••¢ȱŒ‘Š›ŠŒŽ›’£Žȱ‹¢ȱ

‘ŽȱžœŽȱ˜ȱŠȱœ’—•Žȱ•˜’Œȱ–Š›’¡ȱǻŒǯȱǽŗŝǾǼǰȱ‹žȱŠŒŒ˜›’—ȱ˜ȱŸ›˜—ȱŠ—ȱ

ŽŸȱǽřǾǰȱ–Š—¢ȱ˜ȱ‘Ž–ȱ‘ŠŸŽȱ˜—•¢ȱ’—ę—’ŽȱŒ‘Š›ŠŒŽ›’œ’Œȱ–Š›’ŒŽœȱŠ—ȱ then such matrices do not provide a good decision procedure for these

•˜’Œœǯȱ‘Žȱ–Š›’ŒŽœȱŠ••˜ ȱ˜ȱ›Ž™•ŠŒŽǰȱ’—ȱ–Š—¢ȱŒŠœŽœǰȱŠ—ȱ’—ę—’ŽȱŒ‘Š-

›ŠŒŽ›’œ’Œȱ–Š›’¡ȱǻ˜›ȱŠȱ’ŸŽ—ȱ™›˜™˜œ’’˜—Š•ȱ•˜’ŒǼȱ‹¢ȱŠȱꗒŽȱŒ‘Š›ŠŒŽ- ristic Nmatrix and thus obtain metaproperties such as, for example,

ŽŒ’Š‹’•’¢ǯȱ—˜‘Ž›ȱ™›˜‹•Ž–ȱ‘Šȱ–˜’ŸŠŽȱŸ›˜—ȱŠ—ȱ‘’œȱŒ˜•Š‹˜›Š-

˜›œȱ˜ȱ’—›˜žŒŽȱ—˜—ȬŽŽ›–’—’œ–ȱǻŒǯȱǽŚǾǼȱ’œȱ‘ŽȱŠŒȱ‘Šȱ‘Žȱ™›’—Œ’™•Žȱ of truth-funcionality^Ř, inherent to the matrix semantics in general and

˜ȱŒ•Šœœ’ŒŠ•ȱ•˜’Œȱ’—ȱ™Š›’Œž•Š›ǰȱŒ˜—Ě’Œœȱ ’‘ȱ‘Žȱ’—˜›–Š’˜—ȱ™›ŽœŽ—ȱ in the “real world”, which sometimes may be incomplete, inaccurate

Š—Ȧ˜›ȱ’—Œ˜—œ’œŽ—ǯȱ‘žœǰȱŸ›˜—ȱŠ—ȱ‘’œȱŒ˜••Š‹˜›Š˜›œȱ™›˜™˜œŽȱ‘Žȱ use of non-determinism (by means of Nmatrices) in order to weaken the principle of truth-funcionality as a solution to this problem.

•‘˜ž‘ȱ–Š›’ŒŽœȱ‘ŠŸŽȱœ‘˜ —ȱ‘Ž’›ȱžœŽž•—Žœœȱ’—ȱ–Š—¢ȱŽ¡Š–-

™•Žœǰȱ™›˜Ÿ’’—ȱŠȱꗒŠ›¢ȱǻŠ—ȱ‘žœȱŽŒ’Š‹•ŽǼȱœŽ–Š—’Œœȱ˜›ȱ•˜’Œȱ ’-

‘˜žȱ Šȱ ›ž‘Ȭž—Œ’˜—Š•ȱ œŽ–Š—’Œœǰȱ œžŒ‘ȱ Šœȱ œ˜–Žȱ ˜’Œœȱ ˜ȱ ˜›–Š•ȱ

—Œ˜—œ’œŽ—Œ¢ȱȬȱœȱǻŒǯȱǽŚǾǼȱŠ—ȱŒŽ›Š’—ȱ–˜Š•ȱ•˜’ŒœȱǻŒǯȱǽŗřǾǼǰȱŠȱœ’œ- tematic and rigorous study of the algebraic properties of Nmatrices is still missing in the literature. That is, the theory of Nmatrices has not yet been fully developed, from the point of view of its formal proper- ties and expressive power.

ŗȱ ’’˜—Š•ȱ’—˜›–Š’˜—ȱŠ‹˜žȱ•˜’Œȱ–Š›’ŒŽœȱŒŠ—ȱ‹Žȱ˜ž—ȱŠȱǽŗŞǾǰȱǽşǾǰȱǽŗŝǾǰȱǽŗŚǾȱȱŽȱǽŗśǾȱǯ

Ř Principle in which the truth-value of a formula is determined functionally by the truth-value of its immediate sub-formulas.

řŘş Besides the applications to Computer Science mentioned above, there are few studies on non-deterministic algebras from the perspecti-

ŸŽȱ˜ȱ‘Žȱ’œŒ’™•’—Žȱ˜ȱ—’ŸŽ›œŠ•ȱ•Ž‹›Šǯȱ‘ŽȱŽ—Ž›Š•’£Š’˜—ȱ˜ȱ—˜’˜—œȱ

œžŒ‘ȱ Šœȱ ž•›Šȱ ™›˜žŒœǰȱ ›ŽžŒŽȱ –Š›’ŒŽœȱ Š—ȱ ‘Žȱ Ž’‹—’£ȱ ˜™Ž›Š˜›ǰȱ among others, was not studied with full detail in the non-deterministic context. Thus, in this initial paper we propose the formal study of the theory of Nmatrices from the point of view of universal algebra, with the aim of establishing their potential applications in the realm of al- gebraic semantics.

—ȱ™Š›’Œž•Š›ǰȱ Žȱ ’••ȱ˜Œžœȱ˜ž›ȱŽě˜›œȱ’—ȱ‘Žȱ–Ž‘˜˜•˜¢ȱ›˜–ȱ

‹œ›ŠŒȱ•Ž‹›Š’Œȱ˜’Œȱǻǰȱ’—ȱœ‘˜›Ǽǰȱ’—Šžž›ŠŽȱ‹¢ȱǯȱ•˜”ȱŠ—ȱ ǯȱ’˜££’ȱǻœŽŽȱǽŝǾǰȱǽŞǾǰȱǽŜǾǼǰȱŽ¡Ž—’—ȱŽŒ‘—’šžŽœȱ’—Ÿ˜•Ÿ’—ȱžœžŠ•ȱ–Š›’- ces for the more general context of Nmatrices. Thus, many of the known results in the literature on the application of the theory of logic matrices ǻ–˜œȱ˜ȱ‘ŽœŽȱ›Žœž•œȱŒŠ—ȱ‹Žȱ˜ž—ȱ’—ȱǽŗŚǾȱŠ—ȱǽŘŗǾǼȱŒŠ—ȱ‹ŽȱŠ™™•’Žȱ˜ȱ

˜‘Ž›ȱ•˜’Œœȱ‘Šȱ˜ȱ—˜ȱ‘ŠŸŽȱŠȱŒ‘Š›ŠŒŽ›’£Š’˜—ȱ‹¢ȱꗒŽȱ–Š›’ŒŽœǯ This paper contains the initial notions and results developed in what we call Non-deterministic universal algebra, which is basically a

‘Ž˜›¢ȱŽœ’—Žȱ˜ȱŠ—Š•¢£Žȱ›˜–ȱŠȱŸŽ›¢ȱŽ—Ž›Š•ȱ™Ž›œ™ŽŒ’ŸŽȱ‘ŽȱžœžŠ•ȱ concepts and results in universal algebra in order to adapt them to the non-deterministic context.

2

ǯȱљђњђћѡюџѦȱѐќћѐђѝѡѠȱіћȱћіѣђџѠюљȱљєђяџю

In this section we present some common results in universal al-

Ž‹›Šǰȱ›Žšž’›Žȱ˜›ȱ‘ŽȱŽŸŽ•˜™–Ž—ȱ˜ȱŠȱ‘Ž˜›¢ȱ˜ȱ—˜—ȬŽŽ›–’—’œ’Œȱ universal algebra.

'H¿QLWLRQ 6LJQDWXUH A signature Ȉ LV D IDPLO\ ^6n: nԳ} ZKHUHHDFK6 nLVDVHWRIn-ary connectivesVXFKWKDWLIQPWKHQ6n

ˆ6_P = ‡7KHHOHPHQWVRI60DUHFDOOHGconstants. The domainRIȈ LVWKHVHW

|6| = ׫6 ^FF6nIRUVRPHQt 0}.

'H¿QLWLRQ$OJHEUD/HWȈEHDVLJQDWXUH$Qalgebra$IRUȈ LVDSDLU¢$CV²ZKHUH$LVDQRQHPSW\VHWWKHdomainRI$DQGCVLV

řřŖ

DIXQFWLRQWKDWDVVLJQVIRUHYHU\Q•DQGF6nDQRSHUDWLRQCVF Aⁿo$LQ$

:HZLOOXVHWKURXJKRXWWKHWH[WWKHH[SUHVVLRQ„$⁺WRGHQRWH WKHVHW„$^‡`RIDOOWKHQRQHPSW\VXEVHWVRIWKHVHW$$OVRZH ZLOORIWHQLGHQWLI\RQHVLJQDWXUHȈZLWKLWVGRPDLQ_Ȉ_LIWKHODWWHULV¿QLWH DQGLIWKHDULW\RIWKHFRQQHFWLYHVDUHREYLRXVLQWKHFRQWH[W

'H¿QLWLRQ)RUPXODV/HWȈEHDVLJQDWXUHDQGOHWȄEHD FRXQWDEOHVHW^[_PPt`RIV\PEROVFDOOHGvariables7KHDOJHEUD IUHHO\JHQHUDWHGE\ȈIURPȄZLOOEHGHQRWHGE\/ȈȄ7KHHOHPHQWV RI/ȈȄDUHFDOOHGformulasRUschema formulasRYHUȈ

)URPQRZRQDQGJLYHQWKHVHWȄRIYDULDEOHVZHRQO\FRQVLGHU VLJQDWXUHVȈVXFKWKDWȄŀ6n = ‡IRUDOOQ•7KHVHWRIYDULDEOHVRF FXUULQJLQDIRUPXODij/ȈȄZLOOEHGHQRWHGE\9$5ij

'H¿QLWLRQ/HWȈEHDVLJQDWXUHDQGȄ¶Ȅ:HGHQRWHE\/Ȉ Ȅ¶WKHVXEVHWRI/ȈȄIRUPHGE\WKHVFKHPDIRUPXODVijVXFKWKDW 9$5ijŽȄ¶,QSDUWLFXODULIȄ_n ^[_L”L”Q`IRUQ•WKHQ/ȈȄ_n) LVWKHVXEVHWRI/ȈȄIRUPHGE\WKHVFKHPDIRUPXODVijVXFKWKDW 9$5M) Ž^[0«[n}.

'H¿QLWLRQ7RWDODQGSDUWLDOPXOWLIXQFWLRQV/HW$DQG%EH WZRQRQHPSW\VHWV$total multifunctionJIURP%WR$GHQRWHGE\J

%o_M$LVDIXQFWLRQJ%o „$⁺LQWKHXVXDOVHQVH$IXQFWLRQJ%

o„$LQWXUQFRUUHVSRQGVWRZKDWZHFDOODpartial multifunctionJ IURP%WR$

7KURXJKRXWWKHUHVWRIWKLVWH[WZHRQO\XVHWKHFRQFHSWRIWRWDO PXOWLIXQFWLRQ7KXVDWRWDOPXOWLIXQFWLRQZLOOEHUHIHUUHGWRVLPSO\DVD PXOWLIXQFWLRQ

'H¿QLWLRQ&RPSRVLWLRQRIPXOWLIXQFWLRQV/HW$%DQG&

EHQRWHPSW\VHWVDQGOHWJ₁&o_M%DQGJ₂%o_M$EHWZRPXOWLIXQF WLRQV7KHcomposed multifunctionLVWKHPXOWLIXQFWLRQJ₂ƕJ₁&o_M A JLYHQE\J₂ƕJ₁F ׫^J₂EEJ₁F`IRUHYHU\Fא&

řřŗ 7KHSURRIRIWKHIROORZLQJUHVXOWLVVWUDLJKWIRUZDUG

3URSRVLWLRQ 7KH SDUWLDO RSHUDWLRQ RI FRPSRVLWLRQ EHWZHHQ PXOWLIXQFWLRQVLVDVVRFLDWLYH

3

ǯȱȬюљєђяџюѠȱюћёȱȬќњќњќџѝѕіѠњѠ

In this section we present the formal notions of non-deterministic algebras (or ND-algebras) and of homomorphisms between ND-alge- bras, which are fundamental for the development of non-deterministic universal algebra.

'H¿QLWLRQ1'DOJHEUD/HWȈEHDJLYHQVLJQDWXUH$ND- DOJHEUD$RYHUȈLVDSDLU¢$V²ZKHUH$LVDQRQHPSW\VHWWKHdo- mainRI$DQGVLVDIXQFWLRQWKDWDVVLJQVWRHDFKQ•DQGFאȈnD PXOWLIXQFWLRQVF$ⁿo$LQ$VXFKWKDWıFFRUUHVSRQGVWRDQXQLWDU\

VHW$LIFאȈ₀.

:HZLOOZULWHKHQFHIRUZDUGIRUVLPSOLFLW\F^ALQVWHDGRIVF,IFא ȈWKHRQO\HOHPHQWRIF^A ZLOOEHGHQRWHGE\F_AWKDWLVF^A ^F_A`7KURXJK RXWWKLVWH[WZHFDQZULWHƗWRGHQRWHDQ\QWXSOHD₁D_nRIHOHPHQWV LQ$7KDWLVƗEHORQJVWRWKH&DUWHVLDQSURGXFW$ⁿ.

$YURQLQSDQGS@SUHVHQWVWZRQRQGHWHUPLQLVWLFPD WULFHVRU1PDWULFHV0^%DQG0^%WKDWVHPDQWLFDOO\FKDUDFWHUL]HWKH ORJLFDOV\VWHP%ZKLFKLVNQRZQLQOLWHUDWXUHDVPE&RQHRIWKH/RJLFV RI)RUPDO,QFRQVLVWHQF\/),¶V7KHVH1PDWULFHVZLOOEHSUHVHQWHGLQ WKHIROORZLQJWZRH[DPSOHVDQGVXEVHTXHQWO\DQDO\]HGLQWKHOLJKWRI WKHFRQFHSWVLQWURGXFHGDORQJZLWKRWKHU1PDWULFHVLQWURGXFHGLQWKH OLWHUDWXUH

([DPSOH/HWȈ ^š›o™ל`DQGOHW0 = ¢A'2²EH WKH1PDWUL[RYHUȈVXFKWKDW

_ś = {t, t_I, I, f, f_I};

D_ś = {t, t_I, I};

řȱ—›˜žŒŽȱ‹¢ȱǯȱŠ›—’Ž••’ȱŠ—ȱǯȱŠ›Œ˜œȱ’—ȱǽŗŗǾǰȱŠ—ȱ‘Ž›ŽŠŽ›ȱœž’Žȱȱ’—ȱŽŠ’•ȱ’—ȱǽŗŖǾǯ

řřŘ

˜›ȱŽŠŒ‘ȱŒ˜——ŽŒ’ŸŽȱŒǰȱ‘Žȱ–ž•’ž—Œ’˜—ȱ_ś(c) = c^Aśȱ’œȱŽę—Žȱ‹¢ȱ

‘Žȱ˜••˜ ’—ȱŠ‹•Žœȱǻ‘Ž›Žȱȱƽȱǿǰȱ_I}).

ש ^A5 _{W W}

, , I I_,

W ' ' ' ' '

W_, ' ' ' ' '

, ' ' ' ' '

I ' ' ' F F

I_, ' ' ' F F

š ^A W W_, , F I_,

W ' ' ' F F

W_, ' ' ' F F

, ' ' ' F F

I F F F F F

I_, F F F F F

o^A W W_, , I I_,

W ' ' ' F F

W_, ' ' ' F F

, ' ' ' F F

I ' ' ' ' '

I_, ' ' ' ' '

™ ^A

T F

W_, F

, '

F '

I_, '

ל ^A

T '

W_, F

, F

F '

I_, F

řřř

&OHDUO\0LQGXFHVD1'DOJHEUD$ = ¢AV²RYHUȈVXFKWKDWV 2.

([DPSOH/HWȈ ^š›o™ל`DQGOHW0 = ¢A'2²EH WKH1PDWUL[VXFKWKDW

_ř = {t’, I’, f ‘ };

D_ř = {t’, I’};

˜›ȱŽŠŒ‘ȱŒ˜——ŽŒ’ŸŽȱŒǰȱ‘Žȱ–ž•’ž—Œ’˜—ȱ_ř(c) = c^Ařȱ’œȱŽę—Žȱ‹¢ȱ the following tables.

› ^A W¶ ,¶ I¶

W¶ ' ' '

,¶ ' ' '

Iµ ' ' ^Iµ`

š ^A Wµ ,µ Iµ

Wµ ' ' ^Iµ`

,µ ' ' ^Iµ`

Iµ ^Iµ` ^Iµ` ^Iµ`

o ^A Wµ ,µ Iµ

Wµ ' ' ^Iµ`

,µ ' ' ^Iµ`

Iµ ' ' '

™ ^A Wµ ^Iµ`

,µ '

Iµ '

ל ^A

Wµ A

,µ ^Iµ`

Iµ A

řřŚ

/HW$ $ı!VXFKWKDWV 27KXV$LVDQ1'DOJHEUD RYHUȈ

([DPSOH/HWȈ ^š›o™ל`DQGOHW0¶ = ¢A''¶2¶²EH WKH1PDWUL[VXFKWKDW

Ȃ_ř = {t’, t’_I, I’, f ‘, f ‘_I};

D’_ř = {t’, I’};

˜›ȱŽŠŒ‘ȱŒ˜——ŽŒ’ŸŽȱŒǰȱ‘Žȱ–ž•’ž—Œ’˜—ȱȂ_ř(c) = c^AȂřȱ’œȱŽę—Žȱ‹¢ȱ

‘Žȱ˜••˜ ’—ȱŠ‹•Žœȱǻ‘Ž›ŽȱȱȁȱƽȱǿȱȁȀǼǯ

› ^A¶ W¶ W¶_, ,¶ Iµ Iµ_, W¶ '¶ '¶ '¶ '¶ '¶

W¶_, '¶ '¶ '¶ '¶ '¶

,¶ '¶ '¶ '¶ '¶ '¶

Iµ '¶ '¶ '¶ )µ )µ

Iµ_, '¶ '¶ '¶ )µ )µ

š ^A¶ W¶ W¶_, ,¶ Iµ Iµ_,

W¶ '¶ '¶ '¶ )µ )µ

W¶_, '¶ '¶ '¶ )µ )µ

,¶ '¶ '¶ '¶ )µ )µ

Iµ )µ )µ )µ )µ )µ

Iµ_, )µ )µ )µ )µ )µ

o ^A¶ W¶ W¶_, ,¶ Iµ Iµ_, W¶ '¶ '¶ '¶ )µ )µ W¶_, '¶ '¶ '¶ )µ )µ ,¶ '¶ '¶ '¶ )µ )µ Iµ '¶ '¶ '¶ '¶ '¶

Iµ_, '¶ '¶ '¶ '¶ '¶

™^$’

W¶ )µ

W¶_, )µ ,¶ '¶

Iµ '¶

Iµ_, '¶

řřś ל ^A¶

W¶ ^W¶,¶Iµ`

W¶_, ^W¶,¶Iµ`

,¶ )µ

Iµ ^W¶,¶Iµ`

Iµ_, ^W¶,¶Iµ`

&OHDUO\0¶LQGXFHVD1'DOJHEUD$¶ = ¢A'V'²RYHUȈVXFKWKDW ı 2¶.

'H¿QLWLRQ+RPRPRUSKLVPRI1'DOJHEUDV/HW$ ¢$V² DQG% ¢%V'²EHWZR1'DOJHEUDVRYHUDVLJQDWXUHȈ$homomor- phism h: A o%RI1'DOJHEUDVRYHUȈLVDIXQFWLRQK$ื%VXFKWKDW IRUDOOQ•FאȈ_nDQGD₁D_n א$

K>F^AD₁D_n@ŽF^BKD₁«KD_n)).

,QSDUWLFXODUKF_A F_BLIFאȈ₀.⁴

1RWDWLRQ:HZLOOXVHWKHEUDFNHWVµ¶>µ¶DQGµ¶@¶¶WRGLIIHUHQWLDWH ZKHQDIXQFWLRQLVDSSOLHGRQDVHWRIZKHQLWLVDSSOLHGWRDQHOHPHQW RILWVGRPDLQ

([DPSOH/HW$ = ¢Aı²DQG$ = ¢Aı²EHWKH1'DOJHEUDV LQWURGXFHGLQH[DPSOHVDQGUHVSHFWLYHO\/HWK$ĺ$EHDIXQF WLRQVXFKWKDWKW KW_, W¶KW_, ,¶DQGKI KIµ Iµ&OHDUO\

K>'@ 'DQGK) ^Iµ`

KGH¿QHVDKRPRPRUSKLVPK$ĺ$.

2QWKHRWKHUKDQGWKHIXQFWLRQK¶$ĺ$VXFKWKDWK¶W¶ ,K¶

,¶ I_,DQGK¶Iµ K¶Iµ W_,GRHVQRWGH¿QHDKRPRPRUSKLVPK¶$

ĺ$.

+HQFHIRUZDUGDQGZKHQWKHUHLVQRFKDQFHRIFRQIXVLRQZHDV VXPHWKDWWKH1'DOJHEUDVDUHGH¿QHGRYHUD¿[HGVLJQDWXUHȈ

›˜™˜œ’’˜—ȱŗŖDZ /HWȈEHDVLJQDWXUH7KH1'DOJHEUDVRYHUȈ WRJHWKHUZLWKWKHLUKRPRPRUSKLVPVIRUPDFDWHJRU\ZKLFKZLOOEHFDO OHG1'Ȉ

Ś Remember that , if c  6Ŗ, we write V(c) = c^A = {c_A}.

řřŜ

7KHSURRIRIWKLVIDFWLVHDV\LWLVHQRXJKWRVKRZWKDWWKHXVXDO FRPSRVLWLRQRIIXQFWLRQVSURGXFHVDKRPRPRUSKLVPDQGWKDWWKHLGHQ WLW\PDSVSURGXFHWKHLGHQWLW\KRPRPRUSKLVPV

'H¿QLWLRQ)XOOKRPRPRUSKLVPRI1'DOJHEUDV/HW$

¢$V²DQG% ¢%V'²EHWZR1'DOJHEUDVRYHUDVLJQDWXUHȈ$full homomorphism h: A o%RI1'DOJHEUDVRYHUȈLVDIXQFWLRQK$o%

VXFKWKDWKLVDKRPRPRUSKLVPDQGIRUDOOQ!FאȈ_nDQGD₁D_n א$ F^BKD₁«KD_n)) ŽK>F^AD₁D_n@

7KDWLVKLVIXOOKRPRPRUSKLVPLIDQGRQO\LI K>F^AD₁D_n@ F^BKD₁«KD_n)).

IRUDOOFאȈ_nDQGD₁D_nא$ZLWKQ!

4

ǯȱѢяȬȬюљєђяџюѠȱюћёȱѢяȬȬћіѣђџѠђѠǯ

1RZZHDQDO\]HWKHQRWLRQRIVXE1'DOJHEUDIXQGDPHQWDOWR RXURYHUDOOVWXG\RI1'DOJHEUDV

'H¿QLWLRQ 6XE1'DOJHEUD /HW$ ¢$V² DQG % ¢%

V'²EHWZR1'DOJHEUDVRYHUȈVXFKWKDW%ك$:HVD\WKDW%LVDsub- -ND-algebraRI$RYHUȈGHQRWHGE\%Ž$LIIRUHYHU\Q•FאȈ_nDQG E₁E_nא%F^BE₁E_n F^AE₁E_n).

$VZLWKWKHXVXDODOJHEUDVDQRQHPSW\VXEVHWRIWKHGRPDLQRI D1'DOJHEUDJHQHUDWHVDVLQJOHVXE1'DOJHEUD

([DPSOH/HW$ = ¢AV²DQG$ = ¢A'V'²EHWKH1'DOJH EUDVLQWURGXFHGLQH[DPSOHVDQGUHVSHFWLYHO\VXFKWKDW$Ž A'.

%\WKHGH¿QLWLRQRIVDQGV'LVLPPHGLDWHWKDW$LVVXE1' DOJHEUD$¶WKDWLV$Ž A'.

'H¿QLWLRQ6XE1'XQLYHUVH/HW$ ¢$V²EHD1'DOJH EUDRYHUȈ$sub-ND-universeRI$RYHUȈLVDQRQHPSW\VXEVHW%RI

$WKDWLVFORVHGXQGHUWKHRSHUDWLRQVRI$7KDWLVIRUDQ\Q•FאȈ_n DQGE₁E_nא%F^AE₁E_n) Ž%

řřŝ ([DPSOH/HW$ = ¢AV²DQG$ = ¢A'V'²EHWKH1' DOJHEUDVLQWURGXFHGLQH[DPSOHVDQGUHVSHFWLYHO\VXFKWKDW$Ž A'%\WKHGH¿QLWLRQRIV'LVLPPHGLDWHWKDW$LVVXEXQLYHUVHRI$¶.

'H¿QLWLRQ*HQHUDWHGVXE1'XQLYHUVH/HW$ ¢$V²EH D1'DOJHEUDRYHUȈDQG׎;ك A. The sub-universe of A generated by ; over Ȉ,GHQRWHGE\VJ^A_Ȉ;RUVLPSO\VJ;LVGH¿QHGDVIROORZV

VJ; ŀ^%%LVDVXE1'XQLYHUVHRI$RYHU6DQG;Ž%`

1RWHWKDW$LVDDOZD\VDVXE1'XQLYHUVHRI$RYHU6FRQWDLQLQJ

;WKHQ^%%LVDVXE1'XQLYHUVHRI$RYHU6DQG;Ž%`‡7KXV VJ;LVZHOOGH¿QHG

3URSRVLWLRQ7KHVHWVJ;LVDVXE1'XQLYHUVHRI$RYHU6. Proof: 1RWH WKDW VJ[ ك $ DQG VJ;  ‡ EHFDXVH‡ 

;كVJ;/HWQ•FאȈ_nDQGE₁E_n אVJ;/HW%EHDVXE1' XQLYHUVHRI$VXFKWKDW;ك%6LQFHE₁E_nא%WKHQF^AE₁E_n) Ž

%+HQFHF^AE₁E_n) ŽVJ;DQGWKHQVJ;LVDVXE1'XQLYHUVH RI$ ז

$VLQWKHFDVHRIWKHXVXDODOJHEUDVLWLVSRVVLEOHWRJLYHDFRQV WUXFWLYHGH¿QLWLRQRIVJ;

3URSRVLWLRQ/HW$ ¢$V²EHD1'DOJHEUDRQȈDQG׎

;ك$&RQVLGHUWKHIDPLO\^(ⁿ;Q•`RIVXEVHWVGH¿QHGE\LQGXF WLRQDVIROORZV

E^ŖǻǼȱƽȱDzȱ

(ⁿ⁺¹; (ⁿ;׫ ‰^F^AD₁D_NN•FאȈ_NDQGD₁D_N א (ⁿ;`

6RVJ; ‰^(ⁿ;Q•`

7KHSURRILVREWDLQHGE\VKRZLQJVHSDUDWHO\WKDWVJ;Ž‰^(ⁿ; Q•`DQGWKDW‰^(<sup>n</sup>;Q•`ŽVJ;7KH¿UVWKDOILVHDVLO\GRQHE\

GH¿QLWLRQDQGWKHVHFRQGKDOIFDQEHHDVLO\SURYHGE\LQGXFWLRQRQQ 'H¿QLWLRQ6XE1'DOJHEUDJHQHUDWHG/HW$ ¢$V²EH D1'DOJHEUDRYHUȈDQG׎;ك$:HVD\WKDW¢$V²LVgenerated E\;LIVJ; $

řřŞ

ŽȱŒŠ—ȱ—˜ ȱ˜ȱŽę—Žȱ‘Žȱœž‹ȬȬŠ•Ž‹›ŠȱŽ—Ž›ŠŽȱ‹¢ȱŠȱ—˜—- -empty subset of its domain:

3URSRVLWLRQ/HW$ ¢$V²EHD1'DOJHEUDRYHUȈDQG‡

;Ž A. Then SG; ¢VJ;V^;²VXFKWKDWF^SG[D₁D_n F^AD₁ D_nIRUDQ\Q•FאȈ_nDQGD₁D_n אVJ;LVWKHRQO\VXE1' DOJHEUDRIAJHQHUDWHGE\;

Proof: ([LVWHQFH &OHDUO\ SG;Ž $ WKHQ E\ GH¿QLWLRQ VJ;Ž$DQGE\GH¿QLWLRQRISG;F^SG;D₁D_n F^AD₁D_nIRU DQ\Q•FאȈ_nDQGD₁D_nאVJ;

8QLTXHQHVV/HW;DQG<EHWZRVHWVVXFKWKDW;<׎;ك A DQG׎<ك A.

$VVXPHWKDWSG; ¢VJ;V^;²ZKHUHF^SG;D₁D_n F^AD₁ D_nIRUDQ\Q•DQGFאȈ_nLVDVXE1'DOJHEUDRI$JHQHUDWHGE\;

DQGSG< ¢VJ<V^<²VXFKWKDWF^SG<E₁E_n F^AE₁E_nIRUDQ\

Q•FאȈ_nDQGE₁E_nאVJ<LVDVXE1'DOJHEUDRI$JHQHUDWHG E\<&OHDUO\VJ; VJ<DQGF^SG;D₁D_n F^AD₁D_n F^AE₁ E_n F^SG<E₁E_nWKXV¢VJ;V^;² = ¢VJ<V^<².

1RZZHZLOOSURYHWKDWLIK$o%LVDKRPRPRUSKLVPRI1' DOJHEUDVWKHQWKHLPDJHE\K;RISG;LVFRQWDLQHGLQSGK>;@

/HPPD/HW$ ¢$V²DQG% ¢%V'²EHWZR1'DOJHEUDV RYHUȈ‡;ك$DQGOHWK$o%EHDKRPRPRUSKLVPRI1'DOJH EUDV,I(ⁿ;DQG(ⁿK>;@DUHGH¿QHGLQGXFWLYHO\DVLQ3URSRVLWLRQ WKHQK>(Q;@Ž(ⁿK>;@

Proof:7KHSURRILVE\LQGXFWLRQRQQIRUQ•,IQ K>(⁰;@

K>;@ (⁰K>;@6XSSRVHWKDWK>(ⁿ;@Ž(ⁿK>;@WKHQ K>(ⁿ⁺¹;@

K>(ⁿ;׫‰^F^AD₁D_NN•FאȈ_NDQGD₁D_N(ⁿ;`@

K>(ⁿ;@׫ h[‰^F^AD₁D_NN•FאȈ_NDQGD₁D_N(ⁿ;`@

(ⁿK>;@׫‰^K>F^AD₁D_N@N•FאȈ_NDQGD₁D_N(ⁿ;`

(ⁿK>;@׫‰^F^BKD₁KD_N@N•FאȈ_NDQGKD₁ KD_N)

(ⁿK;`

(ⁿ⁺¹K>;@

řřş 7KHRUHP/HW$ ¢$V²DQG% ¢%V'²EHWZR1'DOJH EUDV׎;Ž$DQGOHWK$o%EHDKRPRPRUSKLVPRI1'DOJH EUDV7KHQ

K>VJ;@ŽVJK>;@

Proof:ȱ¢ȱ›˜™˜œ’’˜—ȱŗŜȱ Žȱ‘ŠŸŽ

K>VJ;@ K>‰^(ⁿ;Q•`@ ‰^K>(<sup>n</sup>;@Q•`

œ’—ȱ‘Žȱ™›ŽŸ’˜žœȱ•Ž––ŠȱŠ—ȱ›˜™˜œ’’˜—ȱŗŜȱ Žȱ‘ŠŸŽȱŠŠ’—

‰^K>(ⁿ;@Q•`Ž‰^(<sup>n</sup>K>;@Q•` VJK>;@

7KHUHIRUHK>VJ;@ŽVJK>;@

'H¿QLWLRQ/HW$ ¢$V²DQG% ¢%V'²EHWZR1'DOJHEUDV RYHUDVLJQDWXUHȈK$o%LVDIXOOKRPRPRUSKLVPRI1'DOJHEUDV RYHUȈDQGOHW$ ¢$¶V''²EHDVXE1'DOJHEUD RI$7KHQK$¶

¢K>$@V^KA¶²LVWKHVXE1'DOJHEUDVXFKWKDWIRUDOOQ•FאȈ_nDQG E₁E_nאK>$¶@

F^KA¶E₁E_n) = ‰^K>F^A¶D¶₁D¶_n@KD¶_L E_LIRU”L”Q`

&RUROODU\/HW$ ¢$V²DQG% ¢%V'²EHWZR1'DOJHEUDV RYHUDVLJQDWXUHȈK$o%LVDIXOOKRPRPRUSKLVPRI1'DOJHEUDV DQG‡;ك$7KHQWKHLPDJHE\KRISG;LVDVXE1'DOJHEUDRI SGK>;@

Proof:,WLVFOHDUWKDW7KHRUHPLVVWLOOYDOLGZKHQKLVDIXOOKR PRPRUSKLVPWKXV>VJ;@كVJK>;@DQGIRUDQ\Q•F6nDQGE₁

«E_nK>VJ;@ZHKDYHWKDW

F^KSG;E₁E_n) = ‰^K>F ^SG;D₁D_n@KD_L E_LIRU”L”Q`

‰^K>F^AD₁D_n@KD_L E_LIRU”L”Q`

‰^F^BKD₁KD_nKD_L E_LIRU”L”Q`

F^BE₁E_n F^SGK>;@E₁E_n).

5

ǯȱџќёѢѐѡѠȱќѓȱȬљєђяџюѠǯ

˜ ǰȱ ŽȱŠ—Š•¢£Žȱ‘ŽȱŽę—’’˜—ȱ˜ȱ™›˜žŒœȱ’—ȱ‘ŽȱŒŠŽ˜›¢ȱ˜ȱ ȬŠ•Ž‹›ŠœǰȱŠŠ™’—ȱ‘ŽȱŒ•Šœœ’ŒȱŽę—’’˜—ȱ˜ȱ™›˜žŒœȱ’ŸŽ—ȱ’—ȱž—’-

řŚŖ

versal algebra. Thus, it will be shown that the category of ND-algebra over a given signature is closed by arbitrary products.

'H¿QLWLRQ1'3URGXFWV/HW$1 = ¢A₁V1²DQG$₂ = ¢A₂V2² EHWZR1'DOJHEUDVRYHU67KHGLUHFW1'SURGXFW$₁u A₂LVWKH1' DOJHEUD¢A₁ u A₂V³²RYHU6VXFKWKDWF^A1uA2¢D₁₁D₂₁²«¢D_1nD_2n²) = F^AD₁₁D_1n) uF^AD₂₁D_2nIRUDQ\D_1j A₁DQGD_2j  A₂ZLWK”M

”Q,QSDUWLFXODULIF60F_A1uA2 = ¢F_A1F_A2².

'H¿QLWLRQFDQRQLFDOSURMHFWLRQV/HW$1DQG$₂EHVHWV7KH IXQFWLRQʌ_L: A₁ × A₂ืA_LGH¿QHGE\ʌ_L¢D₁D₂² D_LIRUHYHU\D₁א A₁DQG D₂א A₂LVFDOOHGWKHith- canonical projectionRI$₁ × A₂IRUL

3URSRVLWLRQ/HW$1 = ¢A₁V1²DQG$₂ = ¢A₂V2²EHWZR1' DOJHEUDVRQ67KHQWKHFDQRQLFDOSURMHFWLRQVʌ_L: A₁ × A₂o A_LL DUHIXOOKRPRPRUSKLVPV

7KHSURRILVLPPHGLDWHIURPWKHGH¿QLWLRQV

3URSRVLWLRQ/HW$1 = ¢A₁V1²DQG$₂ = ¢A₂V2²EHWZR1' DOJHEUDVRYHU6DQGOHWʌ_L: A₁ × A₂ o A_LL EHWKHFDQRQLFDO SURMHFWLRQVRI$₁ × A₂. Then ¢A₁ × A₂¢ʌ₁ʌ₂²²LVWKHSURGXFWRI$₁DQG A₂LQWKHFDWHJRU\1'Ȉ

Proof::HKDYHWRVKRZWKDW¢A₁ × A₂¢ʌ₁ʌ₂²²VDWLV¿HVWKHIRO ORZLQJXQLYHUVDOSURSHUW\LI% ¢%V'²LVD1'DOJHEUDDQGI_L%ืA_L IRUL DUHKRPRPRUSKLVPVWKHQWKHUHLVDXQLTXHKRPRPRUSKLVP K%o A₁×A₂VXFKWKDWI_L ʌ_LלKIRUL 7KXVFRQVLGHUWKHIXQFWLRQ K%o A₁×A₂VXFKWKDWKE ¢I₁EI₂E²IRUDOOEא%

,K>F^BE₁E_n@ ^KEEF^BE₁E_n` ^¢I₁EI₂E²E

F^BE₁E_n)} Ž

I₁>F^BE₁E_n@îI₂>F^BE₁E_n@$VE\K\SRWKHVLVI_L%o A_LIRU L DUHKRPRPRUSKLVPVWKHQI_L>F^BE₁E_n@ŽF^AiI_LE₁I_LE_n)) DQGWKXVK>F^BE₁E_n@ŽF^A1I₁E₁I₁E_nîF^A2I₂E₁I₂E_n)) F^Ai×Ai KE₁KE_n7KHUHIRUHK%o A₁×A₂LVDKRPRPRUSKLVP

,,I_LE ʌ_L¢I₁EI₂E² ʌ_LKEIRUL E\WKHGH¿QLWLRQRIK

řŚŗ ,,,6XSSRVHWKDWWKHUHDUHWZRKRPRPRUSKLVPVK₁%o A₁×A₂ DQGK₂%o A₁×A₂VXFKWKDWI_L ʌ_Lלh_jIRULM 6Rʌ_LK₁E I_LE ʌ_LK₂EIRUL DQGEא%7KHUHIRUHK₁ = h = h₂DQGVRWKHKRPR PRUSKLVPK%o A₁×A₂LVXQLTXH

'H¿QLWLRQJHQHUDOFDQRQLFDOSURMHFWLRQV/HW,EHDVHW DQGOHW$_L)_Lא,EHDIDPLO\RI1'DOJHEUDVRYHUȈ7KHIXQFWLRQʌ_j: –_Lא, A_Lo A_jGH¿QHGE\ʌ_jD DMLVFDOOHGWKHj-th canonical projectionRI

–_Lא, A_L.

'H¿QLWLRQJHQHUDOSURGXFWV/HW,DVHWVXFKWKDWLא,DQG

$_L)_Lא,LVDIDPLO\RI1'DOJHEUDVRQȈ7KHSURGXFWGLUHFW$ –_Lא, A_LLV WKH1'DOJHEUD¢–_Lא, A_Lı^3Ȇ²RQȈVXFKWKDWF^AD₁D_n) = –_Lא,F^AiD_L D_LQIRUDOOFאȈ_nDQGD₁D_n א¢–_Lא, A_L².

3URSRVLWLRQ7KHFDQRQLFDOSURMHFWLRQVʌj: –_Lא, A_Lo A_jDUHIXOO KRPRPRUSKLVPV

3URSRVLWLRQ/HW,EHDVHW$_L)_Lא,LVDIDPLO\RI1'DOJHEUDV RYHUȈDQGOHWʌ_j: –_Lא, A_Lo A_jEHWKHMWKFDQRQLFDOSURMHFWLRQRI–_Lא, A_L. Then ¢–_Lא, A_Lʌ_L)_Lא, ²LVWKHSURGXFWRIWKHIDPLO\$_L)_Lא,LQWKHFDWH JRU\1'Ȉ

6

ǯȱћѡђџѝџђѡюѡіќћȱќѓȱѓќџњѢљюѠȱіћȱȬюљєђяџюѠǯ

,QWKLVVHFWLRQZHGH¿QHWKHFRQFHSWRILQWHUSUHWDWLRQRIIRUPXODV RYHUDVLJQDWXUHȈLQDQ1'DOJHEUDRYHUȈ7RGRWKLVZHPXVWXVH DVVLJQPHQWVZKLFKZLOOLQWHUSUHWWKHVFKHPDYDULDEOHVRFFXUULQJLQWKH IRUPXOD

'H¿QLWLRQ6HOHFWRU/HW$DQG%EHQRQHPSW\VHWVJ%

o_M$LVDPXOWLIXQFWLRQDQG$^%LVWKHVHWRIDOOIXQFWLRQVIURP%WR$$

selectorRIJLVDIXQFWLRQȜ%o$VXFKWKDWȜEאJEIRUDOOEא%/HW 6(/J ^Ȝא A^%ȜLVDVHOHFWRURIJ`

'H¿QLWLRQ1'DVVLJQPHQW/HW$ ¢$V²EHD1'DOJH EUD$ND-assigmentLQ$LVDIXQFWLRQȡȄื A.

řŚŘ

1RWHWKDWDVZHOODVWKHFRQVWDQWVDVVXPHDVLQJOHYDOXHLQ1' DOJHEUDVLQVWHDGRIDPXOWLSOLFLW\RIYDOXHVZHZLOOGH¿QHLQDFRKHUHQW ZD\WKDWWKHYDULDEOHVDUHLQVWDQWLDWHGLQLQGLYLGXDOYDOXHVRIWKHDOJH EUDUDWKHUWKDQEHLQJLQVWDQWLDWHGLQQRQHPSW\VHWVRIHOHPHQWVRIWKH DOJHEUD6RIURPRXUSHUVSHFWLYHWKHQRQGHWHUPLQLVPLQWKH1'DOJH EUDVRQO\DSSHDUVLQWKHFRPSOH[OHYHOWKDWLVZKHQRSHUDWRUVGLIIHUHQW RIWKHFRQVWDQWVDUHHIIHFWLYHO\DSSOLHGWRWKHHOHPHQWVRIWKHDOJHEUD

'H¿QLWLRQ LQWHUSUHWDWLRQ RI IRUPXODV LQ D 1'DOJHEUD /HW$ ¢$V²EHD1'DOJHEUDDQGOHWȡEHD1'DVVLJPHQWLQ$7KH PXOWLIXQFWLRQڄ)^Aȡ/ȈȄo_M$LVWKHinterpretationRIijLQ$E\ȡLVWKH QRQHPSW\VXEVHWij^AȡRI$GH¿QHGE\LQGXFWLRQRQWKHFRPSOH[LW\RIWKH IRUPXODMDVIROORZV

ȟ^Aȡ ^ȡȟ`LIȟאȄ F<sup>Aȡ</sup> ^F<sub>A</sub>`LIFאȈ₀

Fij₁ij_n)^Aȡ = ‰^F^AD₁«D_nD_Lאij_L^$ȡIRU”L”Q`LIQ!F אȈ_nDQGij_Lא/ȈȄIRU”L”Q

1RWDWLRQ,IȡLVDQ1'DVVLJQPHQWLQD1'DOJHEUD$ ¢$V²ij א/ȈȄ_nDQGȡȟ_L D_LZLWK”L”QZHZLOOZULWHij^AD₁D_nLQVWHDG RIij^Aȡ.

7

ǯȱȬѐќћєџѢђћѐђѠȱюћёȱȬѢќѡіђћѡȱљєђяџюѠǯ

7KHFRQFHSWVRIFRQJUXHQFHDQGTXRWLHQWDOJHEUDDUHHVVHQWLDO WRROVLQ%ORNDQG3LJR]]L¶VWKHRU\RIDOJHEUDL]DWLRQRIORJLFDOV\VWHPV$L PLQJIRUSRVVLEOHDSSOLFDWLRQVRI1'DOJHEUDWKHRU\ZLWKLQWKHDOJHEUDLF VHPDQWLFVRIORJLFDOV\VWHPVWKLVVHFWLRQZLOOGLVFXVVWKHGH¿QLWLRQRI FRQJUXHQFHDQGTXRWLHQWDOJHEUDLQWKHFRQWH[WRI1'DOJHEUDV

'H¿QLWLRQ1'&RQJUXHQFH/HW$ ¢$V²EHD1'DOJHEUD RYHUDVLJQDWXUHȈDQGOHWTŽ$î$EHDUHODWLRQLQ$:HVD\WKDWTLV DcongruenceLQ$LIDQGRQO\LI

TLVDQHTXLYDOHQFHUHODWLRQ

IRUDOOQ!FאȈ_nDQGD₁D_nE₁E_nא$LID_LTE_LIRUDOO”,

”QWKHQ

řŚř IRUDOODאF^AD₁D_nWKHUHLVEאF^AE₁E_nVXFKWKDWDTE IRUDOOEאF^AE₁E_nWKHUHLVDאF^AD₁D_nVXFKWKDWETD ([DPSOH/HWȈ ^רשo™ל`$¶WKH1'DOJHEUDLQWURGXFHG LQ([DPSOHDQGOHWT ^¢W_,W¶²¢WW_,²¢I_,Iµ²¢II_,²} ‰^¢DD²א$¶}

Ž$¶î$¶. Then TLVDFRQJUXHQFHLQ$.

3URSRVLWLRQ/HW$ ¢$V²EHD1'DOJHEUDRYHUDVLJQDWXUH ȈDQGOHWT Ž$î$EHDFRQJUXHQFHRQ$7KHQIRUDOOijא/ȈȄ_n) ZLWKQ!DQGIRUDOOD₁«D_nE₁«E_nא AⁿVXFKWKDWD_LTE_LIRU”,

”QWKHIROORZLQJKROGV

IRUDOODאij^AD₁«D_nWKHUHLVEאij^AE₁«E_nVXFKWKDWDTE IRUDOOEאij^AE₁«E_nWKHUHLVDאij^AD₁«D_nVXFKWKDWETD 7KHSURRIFDQEHHDVLO\GRQHE\LQGXFWLRQRQWKHFRPSOH[LW\RIij 'H¿QLWLRQ/HW$ ¢$V²EHD1'DOJHEUDRYHUDVLJQDWXUHȈ DQGOHWTŽ$î$EHDFRQJUXHQFHRQ$7KHND-algebra quotientRI$

E\TGHQRWHGE\$TLVWKH1'DOJHEUDRIXQLYHUVH$TZLWKRSHUDWLRQV F^A/TD₁T«DnT ^DTDF^AD₁«D_n`ZKHUHDTLVWKHHTXLYDOHQ FHFODVVRIDDOVRFDOOHGWKHcongruence classRID

3URSRVLWLRQ,I$ ¢$V²LVD1'DOJHEUDRYHUDVLJQDWXUHȈ DQGTŽ$î$LVDFRQJUXHQFHRQ$WKHQ$TLVLQGHHGD1'DOJHEUD ZKRVHRSHUDWLRQVDUHZHOOGH¿QHG

The proof is straightforward.

8

ǯȱіљѡђџѠǰȱѢљѡџюѓіљѡђџѠȱюћёȱѢљѡџюѝџќёѢѐѡѠǯ

,QWKLVVHFWLRQZHZLOOVKRZXVLQJRXUGH¿QLWLRQRI1'DOJHEUD TXRWLHQWWKDWLWLVSRVVLEOHWRGH¿QHWKHXOWUDSURGXFWWKLVLVWKHUHGX FHGSURGXFWZLWKUHVSHFWWRDQXOWUD¿OWHURIDQ\IDPLO\RI1'DOJHEUDV 3URSRVLWLRQ/HW,EHDVHW8Ž„,DQXOWUD¿OWHURQ,$_L)_Lא, DIDPLO\RI1'DOJHEUDVRYHUȈDQGT₈Ž–_Lא, A_LðGH¿QHGDVIROORZV

řŚŚ

DT₈ELIDQGRQO\LI^Lא,DL EL`א87KHQT₈LVDFRQJUXHQFHRQ WKH1'DOJHEUD$ –_Lא, A_L.

Proof:&OHDUO\T₈LVDQHTXLYDOHQFHUHODWLRQ1RZZHVKRZWKDWT₈ VDWLV¿HVWKHGH¿QLWLRQRI1'FRQJUXHQFH/HWQ!FאȈ_nDQGD₁« D_nE₁«E_n A = –_Lא, A_LVXFKWKDWD_jT₈E_jIRU”M”Q7KHQE\GH¿QL WLRQRIT₈DQGE\WKHSURSHUWLHVRI8ZHKDYHWKDW5 ^LD₁L E₁L D_nL E_nL`8

1RZOHW6 ^LF^ALD₁L«D_nL F^ALE₁L«E_nL`8&OHDUO\

5Ž6WKHUHIRUH68

/HW[F^AD₁«D_n) = –_Lא,F^ALD₁L«D_nLDQGGH¿QH\ A VXFKWKDW\L [LIRUL6DQG\LF^ALE₁L«E_nLLIL66LQFH F^AE₁«E_n) = –_Lא,F^ALE₁L«E_nLWKHQ\F^AE₁«E_n0RUHRYHU 6Ž^L[L \L` 7DQGWKHQ787KHUHIRUH[T₈\

$QDORJRXVO\ZHFDQSURYHWKDWLI\F^AE₁«E_nWKHUHLV[

F^AD₁«D_nVXFKWKDW\T₈[

7KLVVKRZVWKDWT₈LVDFRQJUXHQFHRQWKH1'DOJHEUD$

'H¿QLWLRQ8OWUDSURGXFW/HW,EHDVHW8Ž„,DQXOWUD¿OWHU RQ,$_L)_Lא,DIDPLO\RI1'DOJHEUDVRQȈDQGT₈Ž–_Lא, A_L)². The ultra- product –_Lא, A_L8LVWKH1'DOJHEUDTXRWLHQW–_Lא, A_LT₈.

іћюљȱѐќћѠіёђџюѡіќћѠ

7KHVWXG\RIWKHXVXDOORJLFDOPDWULFHVDQG1PDWULFHVEXWPDLQO\

WKHIXQGDPHQWDOWRROVRIXQLYHUVDODOJHEUDHQDEOHGWKHGHYHORSPHQW RIWKH¿UVWRULJLQDOUHVXOWVLQZKDWZHFDOOnon-deterministic universal algebra.

,QWKLVWKHRU\QRQGHWHUPLQLVWLFDOJHEUDLFVWUXFWXUHVFDOOHGND- -algebras ZHUH LQWURGXFHG ZKRVH QRQGHWHUPLQLVWLF RSHUDWLRQV SUR GXFHQRQHPSW\VHWVRIYDOXHVUDWKHUWKDQLQGLYLGXDOYDOXHV6HYHUDO QRWLRQVDQGEDVLFFRQVWUXFWLRQVIURPXQLYHUVDODOJHEUDZHUHDGDSWHG WRWKHQRQGHWHUPLQLVWLFIUDPHZRUN

&RQFHUQLQJWKHQH[WVWHSVZHZLOOIRFXVRXUHIIRUWVLQWKHPHWKR GRORJ\IURP$EVWUDFW$OJHEUDLF/RJLF$$/LQVKRUWLQDXJXUDWHGE\:

řŚś

%ORNDQG'3LJR]]LVHH>@>@>@H[WHQGLQJWHFKQLTXHVLQYROYLQJ XVXDOPDWULFHVIRUWKHPRUHJHQHUDOFRQWH[WRI1PDWULFHV7KXVPDQ\

RIWKHNQRZQUHVXOWVLQWKHOLWHUDWXUHRQWKHDSSOLFDWLRQRIWKHWKHRU\

RIORJLFPDWULFHVPRVWRIWKHVHUHVXOWVFDQEHIRXQGLQ>@DQG>@

FRXOGEHDSSOLHGWRRWKHUORJLFVWKDWGRQRWKDYHDFKDUDFWHUL]DWLRQE\

¿QLWHPDWULFHV

$FNQRZHGJHPHQWV

7KLVSURMHFWZDVVSRQVRUHGE\)$3(63%UD]LO7KHVHFRQGDX WKRUZDVDOVRVXSSRUWHGE\DUHVHDUFKJUDQWIURP&13T%UD]LO

ђѓђџђћѐђѠ

$YURQDQG%.RQLNRZVND0XOWLYDOXHGFDOFXOLIRUORJLFVEDVHGRQQRQGHWHU PLQLVP/RJLF-RXUQDORIWKH,*3/±

$YURQDQG,/HY&DQRQLFDOSURSRVLWLRQDOJHQW]HQW\SHV\VWHPV,Q3URFHH GLQJV RI WKH )LUVW ,QWHUQDWLRQDO -RLQW &RQIHUHQFH RQ$XWRPDWHG 5HDVRQLQJ ,-&$5¶SDJHV±6SULQJHU9HUODJ/RQGRQ8.

$YURQ DQG , /HY 1RQGHWHUPLQLVWLF PXOWLSOHYDOXHG VWUXFWXUHV - /RJ DQG

&RPSXWYROXPHSDJHV±2[IRUG8QLYHUVLW\3UHVV2[IRUG8.

-XQH

$YURQDQG$=DPDQVN\1RQGHWHUPLQLVWLFVHPDQWLFVIRUORJLFDOV\VWHPV,Q 'RY0*DEED\DQG)UDQ]*XHQWKQHUHGLWRUV+DQGERRNRI3KLORVRSKLFDO/R JLFYROXPHRI+DQGERRNRI3KLORVRSKLFDO/RJLFSDJHV±6SULQJHU 1HWKHUODQGV

$UQRQ$YURQ1RQGHWHUPLQLVWLFPDWULFHVDQGPRGXODUVHPDQWLFVRIUXOHV,Q -HDQ<YHV%H]LDXHGLWRU/RJLFD8QLYHUVDOLVSDJHV±%LUNKlXVHU%D VHO

:-%ORNDQG'3LJR]]L$EVWUDFWDOJHEUDLFORJLFDQGWKHGHGXFWLRQWKHRUHP

%XOORI6\PEROLF/RJLFWRDSSHDU

:-%ORNDQG'3LJR]]L$&KDUDFWHUL]DWLRQRI$OJHEUDL]DEOH/RJLFV,QWHUQDO 5HSRUW8QLY,OOLQRLVDW&KLFDJR

:-%ORNDQG'3LJR]]L$OJHEUDL]DEOHORJLFV0HPRULHVRIWKH$PHULFDQ 0DWKHPDWLFDO6RFLHW\

/%ROFDQG3%RURZLN0DQ\9DOXHG/RJLFV9ROXPH7KHRUHWLFDO)RXQGD WLRQV0DQ\YDOXHG/RJLFV6SULQJHU

řŚŜ

:$&DUQLHOOL0(&RQLJOLRDQG-0DUFRV/RJLFVRI)RUPDO,QFRQVLVWHQF\

,Q ' *DEED\ DQG ) *XHQWKQHU HGLWRUV +DQGERRN RI 3KLORVRSKLFDO /RJLF QGHGLWLRQYROXPHSDJHV±6SULQJHU

:$&DUQLHOOLDQG-0DUFRV$WD[RQRP\RI&V\VWHPV,Q:$&DUQLHOOL 0(&RQLJOLRDQG,0/'¶2WWDYLDQRHGLWRUV3DUDFRQVLVWHQF\WKH/RJLFDO :D\WRWKH,QFRQVLVWHQWYROXPHRI/HFWXUH1RWHVLQ3XUHDQG$SSOLHG 0DWKHPDWLFVSDJHV±1HZ<RUN0DUFHO'HNNHU

.'HQHFNHDQG6/:LVPDWK8QLYHUVDO$OJHEUDDQG$SSOLFDWLRQVLQ7KHRUH WLFDO&RPSXWHU6FLHQFH&KDSPDQDQG+DOO

/)DULxDVGHO&HUUR0(&RQLJOLRDQG103HURQ1PDWULFHVIRUPRGDO ORJLF7RDSSHDU

-0)RQWDQG5-DQVDQD$JHQHUDODOJHEUDLFVHPDQWLFVIRUVHQWHQWLDOOR JLFV,Q/HFWXUH1RWHVLQ/RJLF9RO9,,$VVRFLDWLRQIRU6\PEROLF/RJLF 6*RWWZDOG$7UHDWLVHRQ0DQ\9DOXHG/RJLFV6WXGLHVLQ/RJLFDQG&RPSX WDWLRQ5HVHDUFK6WXGLHV3UHVV

5+DKQOH$GYDQFHGPDQ\YDOXHGORJLFV+DQGERRNRI3KLORVRSKLFDO/RJLF

±

-/RĞDQG56XV]NR5HPDUNVRQVHQWHQWLDOORJLFV7KH-RXUQDORI6\PEROLF /RJLFYROXPHSDJHV±'HF

*0DOLQRZVNL0DQ\9DOXHG/RJLFV2[IRUG/RJLF*XLGHV&ODUHQGRQ3UHVV

6 0HOGDO DQG 0 :DOLFNL 1RQGHWHUPLQLVWLF 2SHUDWRUV LQ $OJHEUDLF )UD PHZRUNV 7HFKQLFDO UHSRUW &6/±75±± 3URJUDP$QDO\VLV DQG 9HUL¿

FDWLRQ*URXS5HSRUW1R&RPSXWHU6\VWHPV/DERUDWRU\'HSDUWPHQWRI (OHFWULFDO(QJLQHHULQJDQG&RPSXWHU6FLHQFH6WDQIRUG8QLYHUVLW\

-%5RVVHUDQG$57XUTXHWWH0DQ\YDOXHGORJLFV6WXGLHVLQORJLFDQGWKH IRXQGDWLRQVRIPDWKHPDWLFV*UHHQZRRG3UHVV

5:yMFLFNL/HFWXUHVRQ3URSRVLWLRQDO&DOFXOL3XE+RXVHRIWKH3ROLVK$FD GHP\RI6FLHQFHV