on monadic algebras

[Jorge Petrucio Viana]

Dear filocalcmaths,

sorry, I lost the path to previous msg on this subject.

So, I am sending a "new" one.

I said:

> Just a small observation: (1) and (2) are about logic of monadic predicates, that is, they only encompass quantification in formulas generated from monadic predicates, those of the form P (x). 

Diethard replied:> NO! Monadic in Monadic Boolean Algebra has nothing to do 
with monadic predicates / functions! 

Maybe, we are talking about different things, because in the small book by Halmos and Givant they explicitly write (page 134):
"The theory of monadic algebras is an algebraic treatment of the logic of propositional functions of one argument with Boolean operations and a single (existential) quantifier. (...) The proper framework for an algebraic treatment of predicate logic is polyadic algebra."
When I gave a look into their systems - many years ago - it came up to my mind that monadic algebras are translatable in (a restricted fragment of) monadic FOL.So, I made the claim above.
best,Petrucio

􏰏􏰐􏰔􏰘􏰐􏰔􏰙 􏰙􏰣􏰠􏰙􏰚􏰛􏰓􏰪􏰞􏰛􏰝􏰔􏰟􏰥􏰛􏰒􏰑􏰒􏰛􏰚􏰛􏰝􏰔􏰟􏰥􏱟􏰞􏰘􏰥􏰔􏰛􏰘􏰠􏰔􏰚􏰘􏰙􏰣􏰘􏰐􏰔􏰝􏰙􏰑􏰞

􏱜􏰰􏰰􏰰

􏰙􏰣􏰥􏰙􏰙􏰒􏰑􏰘􏰑􏰙􏰚􏰛􏰝􏰣􏰴􏰚􏰞􏰘􏰑􏰙􏰚􏰒􏰙􏰣􏰙􏰚􏰔􏱔􏰴􏰠􏰔􏰚􏰘􏱉􏰑􏱚􏱅􏰙􏰙􏰝􏰔􏱑􏰙􏰔􏰥􏰛􏰘􏰑􏰙􏰚􏰒􏱑􏰓􏰛 􏰜􏰜􏰰􏰨􏰜

􏰒􏰑􏰚 􏰝􏰔 􏰔􏱈􏰵􏰒􏰘􏰔􏰚􏰘􏰑􏰭 􏰴􏰛􏰚􏰘􏰑􏰕􏰔􏰥􏰗􏰧􏰚 􏰠 􏰙􏰓􏰔􏱣 􏰥􏰔􏰓􏰑􏰞􏰛􏰘􏰔 􏰝􏰙 􏰑􏰞 􏱑 􏰓 􏰵􏰚 􏰻􏰵􏰡􏰴􏰭 􏰝 􏰛􏰝􏰝

􏰰􏰸􏰺􏱢􏰸􏰜􏰺􏰰􏰢

􏰙􏰣􏰠􏰛􏰘􏱤􏰔􏰠􏰛􏰘􏰑􏰞􏰒􏱉􏰔􏰚􏰔􏰔􏰓􏰘􏰙􏰴􏰒􏰔 􏰥􏰙 􏰙􏰒􏰵􏰘􏰵􏰙􏰚􏰛􏰝􏰣􏰴􏰚􏰞􏰘􏰑􏰙􏰚􏰒􏱉􏰵􏰘􏰐􏰛􏰚 􏰕􏰚􏰵􏰘􏰔􏰚􏰴􏰠􏰟􏰔􏰥

􏰜􏰜􏰢

􏰙􏰣􏱔􏰴􏰠􏰔􏰚􏰘􏰒􏰫􏰓􏰐􏰔􏰚􏰞􏰔􏰑􏰚􏰕􏰚􏰑􏰘􏰔􏰝􏰠􏰛􏰚 􏰴􏰛􏰚􏰘􏰑􏰕􏰔􏰥􏰒􏰙􏰚􏰔􏰣􏰙􏰥􏰔􏰛􏰞􏰐􏱔􏰴􏰠􏰔􏰚􏰘􏰗 􏰰􏰨􏰢􏰢􏱢􏰨􏰰

􏰏􏰐􏰔 􏰥􏰙 􏰔􏰥􏰣􏰥􏱥 􏰔􏱉􏰙􏰥􏱝􏰣􏰙􏰥􏰫 􏰛􏰝􏰔􏰟􏰥􏱟􏰞􏰘􏰥􏰔􏰛􏰘􏰠􏰔􏰚􏰘􏰙􏰣 􏰥􏰔􏰓􏰑􏰞􏰛􏰘􏰔􏰝􏰙 􏰵􏰞􏰵􏰒 􏰙􏰝􏰛􏱦􏱧􏰞

􏰜􏰜􏰰􏰜􏰰􏰜􏰢

􏰭􏰔􏰟􏰥􏰛􏰗􏱨􏰒􏰘􏰴􏰓􏰙􏰣􏱚􏰑􏰒􏰘􏰙􏰑􏰞􏰪􏰒􏰟􏰔􏰙􏰚􏰓􏰘􏰐􏰔􏰒􏰞􏰙􏰔􏰙􏰣􏰛􏰒􏰐􏰙􏰡􏰑􏰚􏰘􏰥􏰙􏰓􏰴􏰞􏰘􏰙 􏰟􏰙􏰙􏱝 􏰰􏰢􏰜􏰢􏰜􏱜 

[Diethard Michaelis]

Hi Petrucio, hi all,

See ch. 48. "Monadic algebras" of the booklet (p.117).
It starts with:
"A monadic algebra is a Boolean algebra A
together with an existential quantifier on A.
Every functional monadic algebra is (as the name implies)
an example of a monadic algebra."
That should hopefully clarify our monadic confusion.

Diethard.